NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.6%

Time: 31.0s

Alternatives: 13

Speedup: 2.0×

• 39.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1e-7)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (/ 1.0 (exp (+ x (* x eps_m)))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-7) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / exp((x + (x * eps_m)))) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1d-7) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = ((1.0d0 / exp((x + (x * eps_m)))) + exp((x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-7) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / Math.exp((x + (x * eps_m)))) + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1e-7:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = ((1.0 / math.exp((x + (x * eps_m)))) + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1e-7)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / exp(Float64(x + Float64(x * eps_m)))) + exp(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 <= 1e-7)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = ((1.0 / exp((x + (x * eps_m)))) + exp((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1e-7], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.9999999999999995e-8

   1. Initial program 61.0%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in eps around 0 25.7%

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

      1. associate-+r+ 65.2%

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

      2. mul-1-neg 65.2%

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

      3. sub-neg 65.2%

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

      4. +-inverses 65.2%

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

      5. distribute-lft-out 65.2%

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

      6. distribute-rgt1-in 65.2%

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

      7. mul-1-neg 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in eps around 0 65.2%

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

   if 9.9999999999999995e-8 < 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 90.4%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in eps around inf 99.0%

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

      1. *-commutative 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 74.7%

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

Alternative 2: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Simplified 65.3%

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

5. Taylor expanded in eps around inf 98.7%

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

6. Final simplification 98.7%

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

Alternative 3: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + x \cdot eps\_m}}}{2}\\ \mathbf{elif}\;x \leq 6.5:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+77}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-284)
   (/ (+ 1.0 (/ 1.0 (exp (+ x (* x eps_m))))) 2.0)
   (if (<= x 6.5)
     (/ (+ (exp (* x eps_m)) (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
     (if (<= x 1.8e+77)
       (* (exp (- x)) (+ x 1.0))
       (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-284) {
		tmp = (1.0 + (1.0 / exp((x + (x * eps_m))))) / 2.0;
	} else if (x <= 6.5) {
		tmp = (exp((x * eps_m)) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.8e+77) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	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-284)) then
        tmp = (1.0d0 + (1.0d0 / exp((x + (x * eps_m))))) / 2.0d0
    else if (x <= 6.5d0) then
        tmp = (exp((x * eps_m)) + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 1.8d+77) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    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-284) {
		tmp = (1.0 + (1.0 / Math.exp((x + (x * eps_m))))) / 2.0;
	} else if (x <= 6.5) {
		tmp = (Math.exp((x * eps_m)) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.8e+77) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-284:
		tmp = (1.0 + (1.0 / math.exp((x + (x * eps_m))))) / 2.0
	elif x <= 6.5:
		tmp = (math.exp((x * eps_m)) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	elif x <= 1.8e+77:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-284)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x + Float64(x * eps_m))))) / 2.0);
	elseif (x <= 6.5)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.8e+77)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-284)
		tmp = (1.0 + (1.0 / exp((x + (x * eps_m))))) / 2.0;
	elseif (x <= 6.5)
		tmp = (exp((x * eps_m)) + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.8e+77)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-284], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.5], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8e+77], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.99999999999999973e-284

   1. Initial program 71.2%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 71.8%

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

   if -4.99999999999999973e-284 < x < 6.5

   1. Initial program 50.3%

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

   3. Simplified 40.8%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around inf 98.4%

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

      1. *-commutative 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in x around 0 83.7%

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

       1. mul-1-neg 83.7%

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

   11. Simplified 83.7%

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

   if 6.5 < x < 1.7999999999999999e77

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in eps around 0 60.7%

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

      1. associate-+r+ 65.5%

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

      2. mul-1-neg 65.5%

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

      3. sub-neg 65.5%

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

      4. +-inverses 65.5%

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

      5. distribute-lft-out 65.5%

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

      6. distribute-rgt1-in 65.5%

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

      7. mul-1-neg 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in eps around 0 65.5%

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

   if 1.7999999999999999e77 < 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 30.9%

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

      1. associate-+r+ 30.9%

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

      2. mul-1-neg 30.9%

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

      3. sub-neg 30.9%

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

      4. +-inverses 30.9%

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

      5. distribute-lft-out 30.9%

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

      6. distribute-rgt1-in 30.9%

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

      7. mul-1-neg 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in x around inf 30.9%

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

      1. rec-exp 30.9%

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

      2. associate-*r/ 30.9%

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

      3. *-rgt-identity 30.9%

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

   10. Simplified 30.9%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 30.9%

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

       2. metadata-eval 30.9%

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

       3. associate-/r/ 30.9%

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

       4. metadata-eval 30.9%

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

       5. rec-exp 30.9%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 70.7%

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

       8. sqr-neg 70.7%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 70.7%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 70.7%

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

   12. Applied egg-rr 70.7%

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

   13. Taylor expanded in x around 0 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} \cdot x \]
   14. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) \cdot x \]

   15. Simplified 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + x \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 6.5:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+77}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -95:\\ \;\;\;\;\frac{-x}{e^{x}}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -95.0)
   (/ (- x) (exp x))
   (if (<= x 1.45)
     (+ 1.0 (* (* x x) (- (* x 0.3333333333333333) 0.5)))
     (if (<= x 2e+77)
       (/ x (exp x))
       (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -95.0) {
		tmp = -x / exp(x);
	} else if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 2e+77) {
		tmp = x / exp(x);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	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 <= (-95.0d0)) then
        tmp = -x / exp(x)
    else if (x <= 1.45d0) then
        tmp = 1.0d0 + ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0))
    else if (x <= 2d+77) then
        tmp = x / exp(x)
    else
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    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 <= -95.0) {
		tmp = -x / Math.exp(x);
	} else if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 2e+77) {
		tmp = x / Math.exp(x);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -95.0:
		tmp = -x / math.exp(x)
	elif x <= 1.45:
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5))
	elif x <= 2e+77:
		tmp = x / math.exp(x)
	else:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -95.0)
		tmp = Float64(Float64(-x) / exp(x));
	elseif (x <= 1.45)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)));
	elseif (x <= 2e+77)
		tmp = Float64(x / exp(x));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -95.0)
		tmp = -x / exp(x);
	elseif (x <= 1.45)
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	elseif (x <= 2e+77)
		tmp = x / exp(x);
	else
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -95.0], N[((-x) / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+77], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -95

   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 0.0%

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

      1. associate-+r+ 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. +-inverses 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. distribute-rgt1-in 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. rec-exp 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. frac-2neg 0.0%

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

       2. div-inv 0.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 0.0%

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

       5. sqr-neg 0.0%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   14. Simplified 100.0%

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

   if -95 < x < 1.44999999999999996

   1. Initial program 54.1%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in eps around 0 27.4%

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

      1. associate-+r+ 74.3%

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

      2. mul-1-neg 74.3%

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

      3. sub-neg 74.3%

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

      4. +-inverses 74.3%

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

      5. distribute-lft-out 74.3%

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

      6. distribute-rgt1-in 74.2%

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

      7. mul-1-neg 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   9. Step-by-step derivation

      1. unpow2 74.3%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 74.3%

       \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   if 1.44999999999999996 < x < 1.99999999999999997e77

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in eps around 0 60.7%

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

      1. associate-+r+ 65.5%

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

      2. mul-1-neg 65.5%

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

      3. sub-neg 65.5%

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

      4. +-inverses 65.5%

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

      5. distribute-lft-out 65.5%

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

      6. distribute-rgt1-in 65.5%

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

      7. mul-1-neg 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 61.7%

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

      1. rec-exp 61.7%

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

      2. associate-*r/ 61.7%

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

      3. *-rgt-identity 61.7%

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

   10. Simplified 61.7%

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

   if 1.99999999999999997e77 < 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 30.9%

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

      1. associate-+r+ 30.9%

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

      2. mul-1-neg 30.9%

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

      3. sub-neg 30.9%

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

      4. +-inverses 30.9%

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

      5. distribute-lft-out 30.9%

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

      6. distribute-rgt1-in 30.9%

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

      7. mul-1-neg 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in x around inf 30.9%

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

      1. rec-exp 30.9%

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

      2. associate-*r/ 30.9%

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

      3. *-rgt-identity 30.9%

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

   10. Simplified 30.9%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 30.9%

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

       2. metadata-eval 30.9%

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

       3. associate-/r/ 30.9%

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

       4. metadata-eval 30.9%

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

       5. rec-exp 30.9%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 70.7%

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

       8. sqr-neg 70.7%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 70.7%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 70.7%

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

   12. Applied egg-rr 70.7%

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

   13. Taylor expanded in x around 0 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} \cdot x \]
   14. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) \cdot x \]

   15. Simplified 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95:\\ \;\;\;\;\frac{-x}{e^{x}}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))
   (if (<= x -1300000.0)
     t_0
     (if (<= x 1.45)
       (+ 1.0 (* (* x x) (- (* x 0.3333333333333333) 0.5)))
       (if (<= x 5e+76) (/ x (exp x)) t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	double tmp;
	if (x <= -1300000.0) {
		tmp = t_0;
	} else if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 5e+76) {
		tmp = x / exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    if (x <= (-1300000.0d0)) then
        tmp = t_0
    else if (x <= 1.45d0) then
        tmp = 1.0d0 + ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0))
    else if (x <= 5d+76) then
        tmp = x / exp(x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	double tmp;
	if (x <= -1300000.0) {
		tmp = t_0;
	} else if (x <= 1.45) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 5e+76) {
		tmp = x / Math.exp(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	tmp = 0
	if x <= -1300000.0:
		tmp = t_0
	elif x <= 1.45:
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5))
	elif x <= 5e+76:
		tmp = x / math.exp(x)
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))))
	tmp = 0.0
	if (x <= -1300000.0)
		tmp = t_0;
	elseif (x <= 1.45)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)));
	elseif (x <= 5e+76)
		tmp = Float64(x / exp(x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	tmp = 0.0;
	if (x <= -1300000.0)
		tmp = t_0;
	elseif (x <= 1.45)
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	elseif (x <= 5e+76)
		tmp = x / exp(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1300000.0], t$95$0, If[LessEqual[x, 1.45], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+76], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e6 or 4.99999999999999991e76 < 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 17.7%

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

      1. associate-+r+ 17.7%

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

      2. mul-1-neg 17.7%

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

      3. sub-neg 17.7%

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

      4. +-inverses 17.7%

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

      5. distribute-lft-out 17.7%

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

      6. distribute-rgt1-in 17.7%

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

      7. mul-1-neg 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. rec-exp 17.7%

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

      2. associate-*r/ 17.7%

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

      3. *-rgt-identity 17.7%

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

   10. Simplified 17.7%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 17.7%

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

       2. metadata-eval 17.7%

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

       3. associate-/r/ 17.7%

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

       4. metadata-eval 17.7%

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

       5. rec-exp 17.7%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 40.5%

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

       8. sqr-neg 40.5%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 40.5%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 41.2%

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

   12. Applied egg-rr 41.2%

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

   13. Taylor expanded in x around 0 72.8%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} \cdot x \]
   14. Step-by-step derivation

       1. *-commutative 72.8%

          \[\leadsto \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) \cdot x \]

   15. Simplified 72.8%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \cdot x \]

   if -1.3e6 < x < 1.44999999999999996

   1. Initial program 54.1%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in eps around 0 27.4%

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

      1. associate-+r+ 74.3%

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

      2. mul-1-neg 74.3%

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

      3. sub-neg 74.3%

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

      4. +-inverses 74.3%

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

      5. distribute-lft-out 74.3%

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

      6. distribute-rgt1-in 74.2%

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

      7. mul-1-neg 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in x around 0 74.3%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   9. Step-by-step derivation

      1. unpow2 74.3%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 74.3%

       \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   if 1.44999999999999996 < x < 4.99999999999999991e76

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in eps around 0 60.7%

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

      1. associate-+r+ 65.5%

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

      2. mul-1-neg 65.5%

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

      3. sub-neg 65.5%

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

      4. +-inverses 65.5%

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

      5. distribute-lft-out 65.5%

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

      6. distribute-rgt1-in 65.5%

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

      7. mul-1-neg 65.5%

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

   7. Simplified 65.5%

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

   8. Taylor expanded in x around inf 61.7%

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

      1. rec-exp 61.7%

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

      2. associate-*r/ 61.7%

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

      3. *-rgt-identity 61.7%

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

   10. Simplified 61.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + x \cdot eps\_m}}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+77}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.5e-286)
   (/ (+ 1.0 (/ 1.0 (exp (+ x (* x eps_m))))) 2.0)
   (if (<= x 1.9e+77)
     (* (exp (- x)) (+ x 1.0))
     (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.5e-286) {
		tmp = (1.0 + (1.0 / exp((x + (x * eps_m))))) / 2.0;
	} else if (x <= 1.9e+77) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-3.5d-286)) then
        tmp = (1.0d0 + (1.0d0 / exp((x + (x * eps_m))))) / 2.0d0
    else if (x <= 1.9d+77) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.5e-286) {
		tmp = (1.0 + (1.0 / Math.exp((x + (x * eps_m))))) / 2.0;
	} else if (x <= 1.9e+77) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3.5e-286:
		tmp = (1.0 + (1.0 / math.exp((x + (x * eps_m))))) / 2.0
	elif x <= 1.9e+77:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3.5e-286)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x + Float64(x * eps_m))))) / 2.0);
	elseif (x <= 1.9e+77)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3.5e-286)
		tmp = (1.0 + (1.0 / exp((x + (x * eps_m))))) / 2.0;
	elseif (x <= 1.9e+77)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3.5e-286], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.9e+77], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999988e-286

   1. Initial program 71.2%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 71.8%

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

   if -3.49999999999999988e-286 < x < 1.9000000000000001e77

   1. Initial program 59.5%

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

   3. Simplified 43.2%

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

   5. Taylor expanded in eps around 0 33.7%

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

      1. associate-+r+ 74.2%

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

      2. mul-1-neg 74.2%

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

      3. sub-neg 74.2%

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

      4. +-inverses 74.2%

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

      5. distribute-lft-out 74.2%

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

      6. distribute-rgt1-in 74.1%

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

      7. mul-1-neg 74.1%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in eps around 0 74.2%

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

   if 1.9000000000000001e77 < 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 30.9%

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

      1. associate-+r+ 30.9%

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

      2. mul-1-neg 30.9%

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

      3. sub-neg 30.9%

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

      4. +-inverses 30.9%

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

      5. distribute-lft-out 30.9%

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

      6. distribute-rgt1-in 30.9%

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

      7. mul-1-neg 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in x around inf 30.9%

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

      1. rec-exp 30.9%

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

      2. associate-*r/ 30.9%

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

      3. *-rgt-identity 30.9%

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

   10. Simplified 30.9%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 30.9%

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

       2. metadata-eval 30.9%

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

       3. associate-/r/ 30.9%

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

       4. metadata-eval 30.9%

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

       5. rec-exp 30.9%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 70.7%

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

       8. sqr-neg 70.7%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 70.7%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 70.7%

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

   12. Applied egg-rr 70.7%

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

   13. Taylor expanded in x around 0 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} \cdot x \]
   14. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) \cdot x \]

   15. Simplified 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-286}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + x \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+77}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{-x}{e^{x}}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -700.0)
   (/ (- x) (exp x))
   (if (<= x 1.85e+77)
     (* (exp (- x)) (+ x 1.0))
     (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -700.0) {
		tmp = -x / exp(x);
	} else if (x <= 1.85e+77) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	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 <= (-700.0d0)) then
        tmp = -x / exp(x)
    else if (x <= 1.85d+77) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    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 <= -700.0) {
		tmp = -x / Math.exp(x);
	} else if (x <= 1.85e+77) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -700.0:
		tmp = -x / math.exp(x)
	elif x <= 1.85e+77:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(-x) / exp(x));
	elseif (x <= 1.85e+77)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -700.0)
		tmp = -x / exp(x);
	elseif (x <= 1.85e+77)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -700.0], N[((-x) / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.85e+77], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -700

   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 0.0%

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

      1. associate-+r+ 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. +-inverses 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. distribute-rgt1-in 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. rec-exp 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. frac-2neg 0.0%

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

       2. div-inv 0.0%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 0.0%

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

       5. sqr-neg 0.0%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. *-rgt-identity 100.0%

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

   14. Simplified 100.0%

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

   if -700 < x < 1.84999999999999997e77

   1. Initial program 58.8%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in eps around 0 31.3%

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

      1. associate-+r+ 73.3%

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

      2. mul-1-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-inverses 73.3%

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

      5. distribute-lft-out 73.3%

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

      6. distribute-rgt1-in 73.2%

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

      7. mul-1-neg 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in eps around 0 73.3%

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

   if 1.84999999999999997e77 < 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 30.9%

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

      1. associate-+r+ 30.9%

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

      2. mul-1-neg 30.9%

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

      3. sub-neg 30.9%

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

      4. +-inverses 30.9%

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

      5. distribute-lft-out 30.9%

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

      6. distribute-rgt1-in 30.9%

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

      7. mul-1-neg 30.9%

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

   7. Simplified 30.9%

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

   8. Taylor expanded in x around inf 30.9%

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

      1. rec-exp 30.9%

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

      2. associate-*r/ 30.9%

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

      3. *-rgt-identity 30.9%

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

   10. Simplified 30.9%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 30.9%

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

       2. metadata-eval 30.9%

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

       3. associate-/r/ 30.9%

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

       4. metadata-eval 30.9%

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

       5. rec-exp 30.9%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 70.7%

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

       8. sqr-neg 70.7%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 70.7%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 70.7%

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

   12. Applied egg-rr 70.7%

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

   13. Taylor expanded in x around 0 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} \cdot x \]
   14. Step-by-step derivation

       1. *-commutative 70.7%

          \[\leadsto \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) \cdot x \]

   15. Simplified 70.7%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{-x}{e^{x}}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{if}\;x \leq -12200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 10^{+78}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))))
   (if (<= x -12200.0)
     t_0
     (if (<= x 650.0)
       (+ 1.0 (* (* x x) (- (* x 0.3333333333333333) 0.5)))
       (if (<= x 1e+78) 0.0 t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	double tmp;
	if (x <= -12200.0) {
		tmp = t_0;
	} else if (x <= 650.0) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 1e+78) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    if (x <= (-12200.0d0)) then
        tmp = t_0
    else if (x <= 650.0d0) then
        tmp = 1.0d0 + ((x * x) * ((x * 0.3333333333333333d0) - 0.5d0))
    else if (x <= 1d+78) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	double tmp;
	if (x <= -12200.0) {
		tmp = t_0;
	} else if (x <= 650.0) {
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	} else if (x <= 1e+78) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	tmp = 0
	if x <= -12200.0:
		tmp = t_0
	elif x <= 650.0:
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5))
	elif x <= 1e+78:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))))
	tmp = 0.0
	if (x <= -12200.0)
		tmp = t_0;
	elseif (x <= 650.0)
		tmp = Float64(1.0 + Float64(Float64(x * x) * Float64(Float64(x * 0.3333333333333333) - 0.5)));
	elseif (x <= 1e+78)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	tmp = 0.0;
	if (x <= -12200.0)
		tmp = t_0;
	elseif (x <= 650.0)
		tmp = 1.0 + ((x * x) * ((x * 0.3333333333333333) - 0.5));
	elseif (x <= 1e+78)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -12200.0], t$95$0, If[LessEqual[x, 650.0], N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+78], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -12200 or 1.00000000000000001e78 < 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 17.7%

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

      1. associate-+r+ 17.7%

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

      2. mul-1-neg 17.7%

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

      3. sub-neg 17.7%

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

      4. +-inverses 17.7%

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

      5. distribute-lft-out 17.7%

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

      6. distribute-rgt1-in 17.7%

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

      7. mul-1-neg 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in x around inf 17.7%

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

      1. rec-exp 17.7%

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

      2. associate-*r/ 17.7%

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

      3. *-rgt-identity 17.7%

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

   10. Simplified 17.7%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 17.7%

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

       2. metadata-eval 17.7%

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

       3. associate-/r/ 17.7%

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

       4. metadata-eval 17.7%

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

       5. rec-exp 17.7%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 40.5%

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

       8. sqr-neg 40.5%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 40.5%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 41.2%

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

   12. Applied egg-rr 41.2%

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

   13. Taylor expanded in x around 0 72.8%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)} \cdot x \]
   14. Step-by-step derivation

       1. *-commutative 72.8%

          \[\leadsto \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot 0.16666666666666666}\right)\right)\right) \cdot x \]

   15. Simplified 72.8%

       \[\leadsto \color{blue}{\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)} \cdot x \]

   if -12200 < x < 650

   1. Initial program 53.7%

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

   3. Simplified 33.6%

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

   5. Taylor expanded in eps around 0 27.3%

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

      1. associate-+r+ 74.4%

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

      2. mul-1-neg 74.4%

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

      3. sub-neg 74.4%

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

      4. +-inverses 74.4%

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

      5. distribute-lft-out 74.4%

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

      6. distribute-rgt1-in 74.4%

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

      7. mul-1-neg 74.4%

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

   7. Simplified 74.4%

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

   8. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   9. Step-by-step derivation

      1. unpow2 73.9%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   10. Applied egg-rr 73.9%

       \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   if 650 < x < 1.00000000000000001e78

   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%

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

   5. Taylor expanded in eps around 0 63.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 63.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 63.7%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 63.7%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 63.7%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 63.7%

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

   7. Simplified 63.7%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12200:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\\ \mathbf{elif}\;x \leq 10^{+78}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 0.00034:\\ \;\;\;\;1 + \left(x \cdot \left(eps\_m + 1\right)\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e+154)
   (* x (+ x 1.0))
   (if (<= x 0.00034)
     (+ 1.0 (* (* x (+ eps_m 1.0)) -0.5))
     (if (<= x 7.8e+102) 0.0 (* x (+ 1.0 (* x (* x 0.5))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = x * (x + 1.0);
	} else if (x <= 0.00034) {
		tmp = 1.0 + ((x * (eps_m + 1.0)) * -0.5);
	} else if (x <= 7.8e+102) {
		tmp = 0.0;
	} else {
		tmp = x * (1.0 + (x * (x * 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 <= (-1.35d+154)) then
        tmp = x * (x + 1.0d0)
    else if (x <= 0.00034d0) then
        tmp = 1.0d0 + ((x * (eps_m + 1.0d0)) * (-0.5d0))
    else if (x <= 7.8d+102) then
        tmp = 0.0d0
    else
        tmp = x * (1.0d0 + (x * (x * 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 <= -1.35e+154) {
		tmp = x * (x + 1.0);
	} else if (x <= 0.00034) {
		tmp = 1.0 + ((x * (eps_m + 1.0)) * -0.5);
	} else if (x <= 7.8e+102) {
		tmp = 0.0;
	} else {
		tmp = x * (1.0 + (x * (x * 0.5)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e+154:
		tmp = x * (x + 1.0)
	elif x <= 0.00034:
		tmp = 1.0 + ((x * (eps_m + 1.0)) * -0.5)
	elif x <= 7.8e+102:
		tmp = 0.0
	else:
		tmp = x * (1.0 + (x * (x * 0.5)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(x * Float64(x + 1.0));
	elseif (x <= 0.00034)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(eps_m + 1.0)) * -0.5));
	elseif (x <= 7.8e+102)
		tmp = 0.0;
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = x * (x + 1.0);
	elseif (x <= 0.00034)
		tmp = 1.0 + ((x * (eps_m + 1.0)) * -0.5);
	elseif (x <= 7.8e+102)
		tmp = 0.0;
	else
		tmp = x * (1.0 + (x * (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e+154], N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00034], N[(1.0 + N[(N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e+102], 0.0, N[(x * N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35000000000000003e154

   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 0.0%

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

      1. associate-+r+ 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. +-inverses 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. distribute-rgt1-in 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. rec-exp 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 0.0%

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

       2. metadata-eval 0.0%

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

       3. associate-/r/ 0.0%

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

       4. metadata-eval 0.0%

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

       5. rec-exp 0.0%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 0.0%

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

       8. sqr-neg 0.0%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 0.0%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 1.6%

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

   12. Applied egg-rr 1.6%

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

   13. Taylor expanded in x around 0 100.0%

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

       1. +-commutative 100.0%

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

   15. Simplified 100.0%

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

   if -1.35000000000000003e154 < x < 3.4e-4

   1. Initial program 58.9%

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

   3. Simplified 48.9%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 80.7%

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

   7. Taylor expanded in x around 0 65.1%

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

   if 3.4e-4 < x < 7.7999999999999997e102

   1. Initial program 96.5%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in eps around 0 50.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 50.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 50.8%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 50.8%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 50.8%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 50.8%

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

   7. Simplified 50.8%

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

   if 7.7999999999999997e102 < 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 30.4%

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

      1. associate-+r+ 30.4%

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

      2. mul-1-neg 30.4%

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

      3. sub-neg 30.4%

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

      4. +-inverses 30.4%

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

      5. distribute-lft-out 30.4%

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

      6. distribute-rgt1-in 30.4%

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

      7. mul-1-neg 30.4%

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

   7. Simplified 30.4%

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

   8. Taylor expanded in x around inf 30.4%

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

      1. rec-exp 30.4%

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

      2. associate-*r/ 30.4%

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

      3. *-rgt-identity 30.4%

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

   10. Simplified 30.4%

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

   11. Taylor expanded in x around 0 71.2%

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

   12. Taylor expanded in x around inf 71.2%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(0.5 \cdot x\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative 71.2%

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

   14. Simplified 71.2%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 0.00034:\\ \;\;\;\;1 + \left(x \cdot \left(\varepsilon + 1\right)\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 560:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.65)
   (* x (+ x 1.0))
   (if (<= x 560.0)
     1.0
     (if (<= x 7.1e+102) 0.0 (* x (+ 1.0 (* x (* x 0.5))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.65) {
		tmp = x * (x + 1.0);
	} else if (x <= 560.0) {
		tmp = 1.0;
	} else if (x <= 7.1e+102) {
		tmp = 0.0;
	} else {
		tmp = x * (1.0 + (x * (x * 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 <= (-1.65d0)) then
        tmp = x * (x + 1.0d0)
    else if (x <= 560.0d0) then
        tmp = 1.0d0
    else if (x <= 7.1d+102) then
        tmp = 0.0d0
    else
        tmp = x * (1.0d0 + (x * (x * 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 <= -1.65) {
		tmp = x * (x + 1.0);
	} else if (x <= 560.0) {
		tmp = 1.0;
	} else if (x <= 7.1e+102) {
		tmp = 0.0;
	} else {
		tmp = x * (1.0 + (x * (x * 0.5)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.65:
		tmp = x * (x + 1.0)
	elif x <= 560.0:
		tmp = 1.0
	elif x <= 7.1e+102:
		tmp = 0.0
	else:
		tmp = x * (1.0 + (x * (x * 0.5)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.65)
		tmp = Float64(x * Float64(x + 1.0));
	elseif (x <= 560.0)
		tmp = 1.0;
	elseif (x <= 7.1e+102)
		tmp = 0.0;
	else
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(x * 0.5))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.65)
		tmp = x * (x + 1.0);
	elseif (x <= 560.0)
		tmp = 1.0;
	elseif (x <= 7.1e+102)
		tmp = 0.0;
	else
		tmp = x * (1.0 + (x * (x * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.65], N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 560.0], 1.0, If[LessEqual[x, 7.1e+102], 0.0, N[(x * N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.6499999999999999

   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 0.0%

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

      1. associate-+r+ 0.0%

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

      2. mul-1-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. +-inverses 0.0%

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

      5. distribute-lft-out 0.0%

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

      6. distribute-rgt1-in 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. rec-exp 0.0%

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

      2. associate-*r/ 0.0%

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

      3. *-rgt-identity 0.0%

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

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 0.0%

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

       2. metadata-eval 0.0%

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

       3. associate-/r/ 0.0%

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

       4. metadata-eval 0.0%

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

       5. rec-exp 0.0%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 0.0%

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

       8. sqr-neg 0.0%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 0.0%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 1.6%

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

   12. Applied egg-rr 1.6%

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

   13. Taylor expanded in x around 0 46.2%

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

       1. +-commutative 46.2%

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

   15. Simplified 46.2%

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

   if -1.6499999999999999 < x < 560

   1. Initial program 53.7%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in eps around inf 97.9%

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

   6. Taylor expanded in x around 0 86.2%

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

   7. Taylor expanded in x around 0 73.6%

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

   if 560 < x < 7.0999999999999998e102

   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%

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

   5. Taylor expanded in eps around 0 56.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 56.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 56.7%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 56.7%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 56.7%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 56.7%

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

   7. Simplified 56.7%

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

   if 7.0999999999999998e102 < 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 30.4%

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

      1. associate-+r+ 30.4%

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

      2. mul-1-neg 30.4%

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

      3. sub-neg 30.4%

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

      4. +-inverses 30.4%

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

      5. distribute-lft-out 30.4%

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

      6. distribute-rgt1-in 30.4%

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

      7. mul-1-neg 30.4%

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

   7. Simplified 30.4%

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

   8. Taylor expanded in x around inf 30.4%

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

      1. rec-exp 30.4%

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

      2. associate-*r/ 30.4%

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

      3. *-rgt-identity 30.4%

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

   10. Simplified 30.4%

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

   11. Taylor expanded in x around 0 71.2%

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

   12. Taylor expanded in x around inf 71.2%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(0.5 \cdot x\right)}\right) \]
   13. Step-by-step derivation

       1. *-commutative 71.2%

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

   14. Simplified 71.2%

       \[\leadsto x \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 560:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.1 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ x 1.0))))
   (if (<= x -1.6) t_0 (if (<= x 650.0) 1.0 (if (<= x 1.8e+154) 0.0 t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -1.6) {
		tmp = t_0;
	} else if (x <= 650.0) {
		tmp = 1.0;
	} else if (x <= 1.8e+154) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x + 1.0d0)
    if (x <= (-1.6d0)) then
        tmp = t_0
    else if (x <= 650.0d0) then
        tmp = 1.0d0
    else if (x <= 1.8d+154) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -1.6) {
		tmp = t_0;
	} else if (x <= 650.0) {
		tmp = 1.0;
	} else if (x <= 1.8e+154) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (x + 1.0)
	tmp = 0
	if x <= -1.6:
		tmp = t_0
	elif x <= 650.0:
		tmp = 1.0
	elif x <= 1.8e+154:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.6)
		tmp = t_0;
	elseif (x <= 650.0)
		tmp = 1.0;
	elseif (x <= 1.8e+154)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (x + 1.0);
	tmp = 0.0;
	if (x <= -1.6)
		tmp = t_0;
	elseif (x <= 650.0)
		tmp = 1.0;
	elseif (x <= 1.8e+154)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6], t$95$0, If[LessEqual[x, 650.0], 1.0, If[LessEqual[x, 1.8e+154], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001 or 1.8e154 < 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 14.4%

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

      1. associate-+r+ 14.4%

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

      2. mul-1-neg 14.4%

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

      3. sub-neg 14.4%

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

      4. +-inverses 14.4%

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

      5. distribute-lft-out 14.4%

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

      6. distribute-rgt1-in 14.4%

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

      7. mul-1-neg 14.4%

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

   7. Simplified 14.4%

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

   8. Taylor expanded in x around inf 14.4%

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

      1. rec-exp 14.4%

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

      2. associate-*r/ 14.4%

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

      3. *-rgt-identity 14.4%

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

   10. Simplified 14.4%

       \[\leadsto \color{blue}{\frac{x}{e^{x}}} \]
   11. Step-by-step derivation

       1. clear-num 14.4%

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

       2. metadata-eval 14.4%

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

       3. associate-/r/ 14.4%

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

       4. metadata-eval 14.4%

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

       5. rec-exp 14.4%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 32.5%

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

       8. sqr-neg 32.5%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \cdot x \]

       9. sqrt-unprod 32.5%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot x \]

       10. add-sqr-sqrt 33.4%

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

   12. Applied egg-rr 33.4%

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

   13. Taylor expanded in x around 0 57.4%

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

       1. +-commutative 57.4%

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

   15. Simplified 57.4%

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

   if -1.6000000000000001 < x < 650

   1. Initial program 53.7%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in eps around inf 97.9%

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

   6. Taylor expanded in x around 0 86.2%

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

   7. Taylor expanded in x around 0 73.6%

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

   if 650 < x < 1.8e154

   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%

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

   5. Taylor expanded in eps around 0 48.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 48.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 48.0%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 48.0%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 48.0%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 48.0%

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

   7. Simplified 48.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.5% 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 62.3%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in x around 0 78.5%

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

   7. Taylor expanded in x around 0 60.6%

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

   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%

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

   5. Taylor expanded in eps around 0 40.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 40.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 40.3%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 40.3%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 40.3%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 40.3%

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

   7. Simplified 40.3%

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

Alternative 13: 15.6% 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 72.0%

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

3. Simplified 65.3%

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

5. Taylor expanded in eps around 0 12.0%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
6. Step-by-step derivation

   1. div-sub 12.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

   2. mul-1-neg 12.0%

      \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

   3. rec-exp 12.0%

      \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

   4. +-inverses 12.3%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   5. metadata-eval 12.3%

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

7. Simplified 12.3%

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

Reproduce

herbie shell --seed 2024143 
(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))