NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 98.8%

Time: 13.3s

Alternatives: 16

Speedup: 1.1×

• 38.6% 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: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

3. Simplified 62.7%

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

5. Taylor expanded in eps around inf 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 65.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-297}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + x \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-297)
   (/ (+ 1.0 (/ 1.0 (exp (+ x (* x eps))))) 2.0)
   (if (<= x 2e+261)
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-297) {
		tmp = (1.0 + (1.0 / exp((x + (x * eps))))) / 2.0;
	} else if (x <= 2e+261) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-297)) then
        tmp = (1.0d0 + (1.0d0 / exp((x + (x * eps))))) / 2.0d0
    else if (x <= 2d+261) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-297) {
		tmp = (1.0 + (1.0 / Math.exp((x + (x * eps))))) / 2.0;
	} else if (x <= 2e+261) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-297:
		tmp = (1.0 + (1.0 / math.exp((x + (x * eps))))) / 2.0
	elif x <= 2e+261:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-297)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x + Float64(x * eps))))) / 2.0);
	elseif (x <= 2e+261)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-297)
		tmp = (1.0 + (1.0 / exp((x + (x * eps))))) / 2.0;
	elseif (x <= 2e+261)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-297], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x + N[(x * eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+261], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000008e-297

   1. Initial program 68.6%

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

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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. Taylor expanded in x around 0 72.9%

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

   7. Taylor expanded in eps around 0 71.3%

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

      1. mul-1-neg 71.3%

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

      2. unsub-neg 71.3%

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

   9. Simplified 71.3%

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

   10. Taylor expanded in x around 0 70.9%

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

   if -2.00000000000000008e-297 < x < 1.9999999999999999e261

   1. Initial program 70.2%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 68.2%

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

   7. Taylor expanded in eps around inf 68.7%

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

      1. *-commutative 68.7%

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

   9. Simplified 68.7%

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

   if 1.9999999999999999e261 < 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 73.2%

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

      1. associate-+r+ 73.2%

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

      2. mul-1-neg 73.2%

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

      3. sub-neg 73.2%

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

      4. +-inverses 73.2%

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

      5. associate-*r* 73.2%

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

      6. distribute-rgt-out 73.2%

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

      7. mul-1-neg 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in eps around 0 73.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-297}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + x \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \frac{-1}{e^{x}}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+259}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -75000000.0)
   (* x (/ -1.0 (exp x)))
   (if (<= x 9.4e+259)
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (-1.0 / exp(x));
	} else if (x <= 9.4e+259) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-75000000.0d0)) then
        tmp = x * ((-1.0d0) / exp(x))
    else if (x <= 9.4d+259) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (-1.0 / Math.exp(x));
	} else if (x <= 9.4e+259) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -75000000.0:
		tmp = x * (-1.0 / math.exp(x))
	elif x <= 9.4e+259:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -75000000.0)
		tmp = Float64(x * Float64(-1.0 / exp(x)));
	elseif (x <= 9.4e+259)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -75000000.0)
		tmp = x * (-1.0 / exp(x));
	elseif (x <= 9.4e+259)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -75000000.0], N[(x * N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.4e+259], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e7

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. neg-mul-1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. mul-1-neg 0.0%

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

      6. mul-1-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. neg-mul-1 100.0%

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

      10. exp-neg 100.0%

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

      11. un-div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -7.5e7 < x < 9.40000000000000015e259

   1. Initial program 63.8%

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

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 74.4%

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

   7. Taylor expanded in eps around inf 74.9%

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

      1. *-commutative 74.9%

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

   9. Simplified 74.9%

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

   if 9.40000000000000015e259 < 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 73.2%

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

      1. associate-+r+ 73.2%

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

      2. mul-1-neg 73.2%

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

      3. sub-neg 73.2%

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

      4. +-inverses 73.2%

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

      5. associate-*r* 73.2%

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

      6. distribute-rgt-out 73.2%

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

      7. mul-1-neg 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in eps around 0 73.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \frac{-1}{e^{x}}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+259}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \frac{-1}{e^{x}}\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(t\_0 \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= x -75000000.0)
     (* x (/ -1.0 (exp x)))
     (if (<= x 1.35)
       (/
        (/
         (*
          eps
          (*
           t_0
           (+ 1.0 (* x (+ (* x (+ 0.5 (* x -0.16666666666666666))) -1.0)))))
         eps)
        2.0)
       (if (<= x 1.65e+83)
         (/ x (exp x))
         (if (<= x 5e+258)
           (/ (/ (* eps (* t_0 (+ 1.0 (* x (+ (* x 0.5) -1.0))))) eps) 2.0)
           0.0))))))

C

double code(double x, double eps) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (-1.0 / exp(x));
	} else if (x <= 1.35) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	} else if (x <= 1.65e+83) {
		tmp = x / exp(x);
	} else if (x <= 5e+258) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    if (x <= (-75000000.0d0)) then
        tmp = x * ((-1.0d0) / exp(x))
    else if (x <= 1.35d0) then
        tmp = ((eps * (t_0 * (1.0d0 + (x * ((x * (0.5d0 + (x * (-0.16666666666666666d0)))) + (-1.0d0)))))) / eps) / 2.0d0
    else if (x <= 1.65d+83) then
        tmp = x / exp(x)
    else if (x <= 5d+258) then
        tmp = ((eps * (t_0 * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0)))))) / eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (-1.0 / Math.exp(x));
	} else if (x <= 1.35) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	} else if (x <= 1.65e+83) {
		tmp = x / Math.exp(x);
	} else if (x <= 5e+258) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if x <= -75000000.0:
		tmp = x * (-1.0 / math.exp(x))
	elif x <= 1.35:
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0
	elif x <= 1.65e+83:
		tmp = x / math.exp(x)
	elif x <= 5e+258:
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (x <= -75000000.0)
		tmp = Float64(x * Float64(-1.0 / exp(x)));
	elseif (x <= 1.35)
		tmp = Float64(Float64(Float64(eps * Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0);
	elseif (x <= 1.65e+83)
		tmp = Float64(x / exp(x));
	elseif (x <= 5e+258)
		tmp = Float64(Float64(Float64(eps * Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))))) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (x <= -75000000.0)
		tmp = x * (-1.0 / exp(x));
	elseif (x <= 1.35)
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	elseif (x <= 1.65e+83)
		tmp = x / exp(x);
	elseif (x <= 5e+258)
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -75000000.0], N[(x * N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35], N[(N[(N[(eps * N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e+83], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+258], N[(N[(N[(eps * N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.5e7

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. neg-mul-1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. mul-1-neg 0.0%

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

      6. mul-1-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. neg-mul-1 100.0%

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

      10. exp-neg 100.0%

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

      11. un-div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -7.5e7 < x < 1.3500000000000001

   1. Initial program 51.5%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in eps around 0 27.1%

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

      1. associate-+r+ 76.1%

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

      2. mul-1-neg 76.1%

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

      3. sub-neg 76.1%

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

      4. +-inverses 76.1%

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

      5. associate-*r* 76.1%

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

      6. distribute-rgt-out 76.7%

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

      7. mul-1-neg 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in x around 0 75.8%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   if 1.3500000000000001 < x < 1.64999999999999992e83

   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 69.3%

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

      1. associate-+r+ 69.3%

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

      2. mul-1-neg 69.3%

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

      3. sub-neg 69.3%

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

      4. +-inverses 69.3%

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

      5. associate-*r* 69.3%

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

      6. distribute-rgt-out 69.3%

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

      7. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in x around inf 69.3%

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

      1. exp-neg 69.3%

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

      2. un-div-inv 69.3%

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

   10. Applied egg-rr 69.3%

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

   if 1.64999999999999992e83 < x < 5e258

   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 37.1%

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

      1. associate-+r+ 37.1%

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

      2. mul-1-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. +-inverses 37.1%

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

      5. associate-*r* 37.1%

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

      6. distribute-rgt-out 37.1%

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

      7. mul-1-neg 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in x around 0 64.5%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in eps around 0 64.5%

      \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \left(\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right) \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]

   if 5e258 < 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 73.2%

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

      1. div-sub 73.2%

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

      2. mul-1-neg 73.2%

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

      3. rec-exp 73.2%

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

      4. +-inverses 73.2%

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

      5. metadata-eval 73.2%

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

   7. Simplified 73.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \frac{-1}{e^{x}}\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+83}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;x \leq -150000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(t\_0 \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= x -150000000.0)
     (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))
     (if (<= x 1.35)
       (/
        (/
         (*
          eps
          (*
           t_0
           (+ 1.0 (* x (+ (* x (+ 0.5 (* x -0.16666666666666666))) -1.0)))))
         eps)
        2.0)
       (if (<= x 5e+82)
         (/ x (exp x))
         (if (<= x 5e+258)
           (/ (/ (* eps (* t_0 (+ 1.0 (* x (+ (* x 0.5) -1.0))))) eps) 2.0)
           0.0))))))

C

double code(double x, double eps) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (x <= -150000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 1.35) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	} else if (x <= 5e+82) {
		tmp = x / exp(x);
	} else if (x <= 5e+258) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    if (x <= (-150000000.0d0)) then
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    else if (x <= 1.35d0) then
        tmp = ((eps * (t_0 * (1.0d0 + (x * ((x * (0.5d0 + (x * (-0.16666666666666666d0)))) + (-1.0d0)))))) / eps) / 2.0d0
    else if (x <= 5d+82) then
        tmp = x / exp(x)
    else if (x <= 5d+258) then
        tmp = ((eps * (t_0 * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0)))))) / eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (x <= -150000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 1.35) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	} else if (x <= 5e+82) {
		tmp = x / Math.exp(x);
	} else if (x <= 5e+258) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if x <= -150000000.0:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	elif x <= 1.35:
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0
	elif x <= 5e+82:
		tmp = x / math.exp(x)
	elif x <= 5e+258:
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (x <= -150000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	elseif (x <= 1.35)
		tmp = Float64(Float64(Float64(eps * Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0);
	elseif (x <= 5e+82)
		tmp = Float64(x / exp(x));
	elseif (x <= 5e+258)
		tmp = Float64(Float64(Float64(eps * Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))))) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (x <= -150000000.0)
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	elseif (x <= 1.35)
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	elseif (x <= 5e+82)
		tmp = x / exp(x);
	elseif (x <= 5e+258)
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -150000000.0], 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, 1.35], N[(N[(N[(eps * N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+82], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e+258], N[(N[(N[(eps * N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.5e8

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. pow1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

   10. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   12. Simplified 1.6%

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

   13. Taylor expanded in x around 0 75.8%

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

       1. *-commutative 75.8%

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

   15. Simplified 75.8%

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

   if -1.5e8 < x < 1.3500000000000001

   1. Initial program 51.5%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in eps around 0 27.1%

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

      1. associate-+r+ 76.1%

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

      2. mul-1-neg 76.1%

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

      3. sub-neg 76.1%

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

      4. +-inverses 76.1%

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

      5. associate-*r* 76.1%

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

      6. distribute-rgt-out 76.7%

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

      7. mul-1-neg 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in x around 0 75.8%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   if 1.3500000000000001 < x < 5.00000000000000015e82

   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 69.3%

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

      1. associate-+r+ 69.3%

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

      2. mul-1-neg 69.3%

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

      3. sub-neg 69.3%

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

      4. +-inverses 69.3%

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

      5. associate-*r* 69.3%

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

      6. distribute-rgt-out 69.3%

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

      7. mul-1-neg 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in x around inf 69.3%

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

      1. exp-neg 69.3%

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

      2. un-div-inv 69.3%

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

   10. Applied egg-rr 69.3%

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

   if 5.00000000000000015e82 < x < 5e258

   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 37.1%

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

      1. associate-+r+ 37.1%

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

      2. mul-1-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. +-inverses 37.1%

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

      5. associate-*r* 37.1%

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

      6. distribute-rgt-out 37.1%

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

      7. mul-1-neg 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in x around 0 64.5%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in eps around 0 64.5%

      \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \left(\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right) \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]

   if 5e258 < 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 73.2%

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

      1. div-sub 73.2%

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

      2. mul-1-neg 73.2%

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

      3. rec-exp 73.2%

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

      4. +-inverses 73.2%

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

      5. metadata-eval 73.2%

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

   7. Simplified 73.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+82}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \frac{-1}{e^{x}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -75000000.0)
   (* x (/ -1.0 (exp x)))
   (if (<= x 3.2e+261) (/ (+ 1.0 (exp (* x eps))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (-1.0 / exp(x));
	} else if (x <= 3.2e+261) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-75000000.0d0)) then
        tmp = x * ((-1.0d0) / exp(x))
    else if (x <= 3.2d+261) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (-1.0 / Math.exp(x));
	} else if (x <= 3.2e+261) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -75000000.0:
		tmp = x * (-1.0 / math.exp(x))
	elif x <= 3.2e+261:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -75000000.0)
		tmp = Float64(x * Float64(-1.0 / exp(x)));
	elseif (x <= 3.2e+261)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -75000000.0)
		tmp = x * (-1.0 / exp(x));
	elseif (x <= 3.2e+261)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -75000000.0], N[(x * N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+261], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e7

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. neg-mul-1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. mul-1-neg 0.0%

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

      6. mul-1-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. neg-mul-1 100.0%

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

      10. exp-neg 100.0%

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

      11. un-div-inv 100.0%

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

   10. Applied egg-rr 100.0%

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

   if -7.5e7 < x < 3.20000000000000021e261

   1. Initial program 63.8%

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

   3. Simplified 53.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 74.4%

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

   7. Taylor expanded in eps around inf 74.9%

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

      1. *-commutative 74.9%

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

   9. Simplified 74.9%

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

   if 3.20000000000000021e261 < 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 73.2%

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

      1. div-sub 73.2%

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

      2. mul-1-neg 73.2%

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

      3. rec-exp 73.2%

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

      4. +-inverses 73.2%

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

      5. metadata-eval 73.2%

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

   7. Simplified 73.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \frac{-1}{e^{x}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + x \cdot 2\\ \mathbf{if}\;x \leq -520000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(t\_0 \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(t\_0 \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* x 2.0))))
   (if (<= x -520000000.0)
     (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))
     (if (<= x 1.6)
       (/
        (/
         (*
          eps
          (*
           t_0
           (+ 1.0 (* x (+ (* x (+ 0.5 (* x -0.16666666666666666))) -1.0)))))
         eps)
        2.0)
       (if (<= x 5.5e+82)
         0.0
         (if (<= x 2e+258)
           (/ (/ (* eps (* t_0 (+ 1.0 (* x (+ (* x 0.5) -1.0))))) eps) 2.0)
           0.0))))))

C

double code(double x, double eps) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (x <= -520000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 1.6) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	} else if (x <= 5.5e+82) {
		tmp = 0.0;
	} else if (x <= 2e+258) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (x * 2.0d0)
    if (x <= (-520000000.0d0)) then
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    else if (x <= 1.6d0) then
        tmp = ((eps * (t_0 * (1.0d0 + (x * ((x * (0.5d0 + (x * (-0.16666666666666666d0)))) + (-1.0d0)))))) / eps) / 2.0d0
    else if (x <= 5.5d+82) then
        tmp = 0.0d0
    else if (x <= 2d+258) then
        tmp = ((eps * (t_0 * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0)))))) / eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 2.0 + (x * 2.0);
	double tmp;
	if (x <= -520000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 1.6) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	} else if (x <= 5.5e+82) {
		tmp = 0.0;
	} else if (x <= 2e+258) {
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 2.0 + (x * 2.0)
	tmp = 0
	if x <= -520000000.0:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	elif x <= 1.6:
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0
	elif x <= 5.5e+82:
		tmp = 0.0
	elif x <= 2e+258:
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(2.0 + Float64(x * 2.0))
	tmp = 0.0
	if (x <= -520000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	elseif (x <= 1.6)
		tmp = Float64(Float64(Float64(eps * Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0);
	elseif (x <= 5.5e+82)
		tmp = 0.0;
	elseif (x <= 2e+258)
		tmp = Float64(Float64(Float64(eps * Float64(t_0 * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))))) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 2.0 + (x * 2.0);
	tmp = 0.0;
	if (x <= -520000000.0)
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	elseif (x <= 1.6)
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0))))) / eps) / 2.0;
	elseif (x <= 5.5e+82)
		tmp = 0.0;
	elseif (x <= 2e+258)
		tmp = ((eps * (t_0 * (1.0 + (x * ((x * 0.5) + -1.0))))) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520000000.0], 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, 1.6], N[(N[(N[(eps * N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.5e+82], 0.0, If[LessEqual[x, 2e+258], N[(N[(N[(eps * N[(t$95$0 * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.2e8

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. pow1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

   10. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   12. Simplified 1.6%

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

   13. Taylor expanded in x around 0 75.8%

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

       1. *-commutative 75.8%

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

   15. Simplified 75.8%

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

   if -5.2e8 < x < 1.6000000000000001

   1. Initial program 51.5%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in eps around 0 27.1%

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

      1. associate-+r+ 76.1%

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

      2. mul-1-neg 76.1%

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

      3. sub-neg 76.1%

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

      4. +-inverses 76.1%

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

      5. associate-*r* 76.1%

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

      6. distribute-rgt-out 76.7%

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

      7. mul-1-neg 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in x around 0 75.8%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   if 1.6000000000000001 < x < 5.49999999999999997e82 or 2.00000000000000011e258 < 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 70.8%

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

      1. div-sub 70.8%

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

      2. mul-1-neg 70.8%

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

      3. rec-exp 70.8%

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

      4. +-inverses 70.8%

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

      5. metadata-eval 70.8%

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

   7. Simplified 70.8%

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

   if 5.49999999999999997e82 < x < 2.00000000000000011e258

   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 37.1%

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

      1. associate-+r+ 37.1%

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

      2. mul-1-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. +-inverses 37.1%

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

      5. associate-*r* 37.1%

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

      6. distribute-rgt-out 37.1%

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

      7. mul-1-neg 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in x around 0 64.5%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in eps around 0 64.5%

      \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \left(\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right) \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+258}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{if}\;x \leq -128000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\frac{t\_0}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+83}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+261}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot t\_0}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 2.0 (* x 2.0)) (+ 1.0 (* x (+ (* x 0.5) -1.0))))))
   (if (<= x -128000000.0)
     (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))
     (if (<= x 500.0)
       (/ t_0 2.0)
       (if (<= x 1.1e+83)
         0.0
         (if (<= x 1e+261) (/ (/ (* eps t_0) eps) 2.0) 0.0))))))

C

double code(double x, double eps) {
	double t_0 = (2.0 + (x * 2.0)) * (1.0 + (x * ((x * 0.5) + -1.0)));
	double tmp;
	if (x <= -128000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 500.0) {
		tmp = t_0 / 2.0;
	} else if (x <= 1.1e+83) {
		tmp = 0.0;
	} else if (x <= 1e+261) {
		tmp = ((eps * t_0) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 + (x * 2.0d0)) * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))
    if (x <= (-128000000.0d0)) then
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    else if (x <= 500.0d0) then
        tmp = t_0 / 2.0d0
    else if (x <= 1.1d+83) then
        tmp = 0.0d0
    else if (x <= 1d+261) then
        tmp = ((eps * t_0) / eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (2.0 + (x * 2.0)) * (1.0 + (x * ((x * 0.5) + -1.0)));
	double tmp;
	if (x <= -128000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 500.0) {
		tmp = t_0 / 2.0;
	} else if (x <= 1.1e+83) {
		tmp = 0.0;
	} else if (x <= 1e+261) {
		tmp = ((eps * t_0) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (2.0 + (x * 2.0)) * (1.0 + (x * ((x * 0.5) + -1.0)))
	tmp = 0
	if x <= -128000000.0:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	elif x <= 500.0:
		tmp = t_0 / 2.0
	elif x <= 1.1e+83:
		tmp = 0.0
	elif x <= 1e+261:
		tmp = ((eps * t_0) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(2.0 + Float64(x * 2.0)) * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))))
	tmp = 0.0
	if (x <= -128000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	elseif (x <= 500.0)
		tmp = Float64(t_0 / 2.0);
	elseif (x <= 1.1e+83)
		tmp = 0.0;
	elseif (x <= 1e+261)
		tmp = Float64(Float64(Float64(eps * t_0) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (2.0 + (x * 2.0)) * (1.0 + (x * ((x * 0.5) + -1.0)));
	tmp = 0.0;
	if (x <= -128000000.0)
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	elseif (x <= 500.0)
		tmp = t_0 / 2.0;
	elseif (x <= 1.1e+83)
		tmp = 0.0;
	elseif (x <= 1e+261)
		tmp = ((eps * t_0) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -128000000.0], 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, 500.0], N[(t$95$0 / 2.0), $MachinePrecision], If[LessEqual[x, 1.1e+83], 0.0, If[LessEqual[x, 1e+261], N[(N[(N[(eps * t$95$0), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.28e8

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. pow1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

   10. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   12. Simplified 1.6%

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

   13. Taylor expanded in x around 0 75.8%

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

       1. *-commutative 75.8%

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

   15. Simplified 75.8%

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

   if -1.28e8 < x < 500

   1. Initial program 51.9%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in eps around 0 26.9%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. associate-*r* 75.6%

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

      6. distribute-rgt-out 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in x around 0 75.3%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in eps around 0 75.3%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right) \cdot \left(2 + 2 \cdot x\right)}}{2} \]

   if 500 < x < 1.09999999999999999e83 or 9.9999999999999993e260 < 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 73.5%

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

      1. div-sub 73.5%

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

      2. mul-1-neg 73.5%

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

      3. rec-exp 73.5%

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

      4. +-inverses 73.5%

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

      5. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   if 1.09999999999999999e83 < x < 9.9999999999999993e260

   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 37.1%

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

      1. associate-+r+ 37.1%

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

      2. mul-1-neg 37.1%

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

      3. sub-neg 37.1%

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

      4. +-inverses 37.1%

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

      5. associate-*r* 37.1%

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

      6. distribute-rgt-out 37.1%

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

      7. mul-1-neg 37.1%

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

   7. Simplified 37.1%

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

   8. Taylor expanded in x around 0 64.5%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in eps around 0 64.5%

      \[\leadsto \frac{\frac{\color{blue}{\varepsilon \cdot \left(\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right) \cdot \left(2 + 2 \cdot x\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -128000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;\frac{\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+83}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+261}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.7% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(x \cdot 0.5 + -1\right)\\ \mathbf{if}\;x \leq -700000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;\frac{\left(2 + x \cdot 2\right) \cdot t\_0}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+257}:\\ \;\;\;\;x \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (+ (* x 0.5) -1.0)))))
   (if (<= x -700000000.0)
     (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))
     (if (<= x 650.0)
       (/ (* (+ 2.0 (* x 2.0)) t_0) 2.0)
       (if (<= x 5e+102) 0.0 (if (<= x 1e+257) (* x t_0) 0.0))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (x * ((x * 0.5) + -1.0));
	double tmp;
	if (x <= -700000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 650.0) {
		tmp = ((2.0 + (x * 2.0)) * t_0) / 2.0;
	} else if (x <= 5e+102) {
		tmp = 0.0;
	} else if (x <= 1e+257) {
		tmp = x * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * ((x * 0.5d0) + (-1.0d0)))
    if (x <= (-700000000.0d0)) then
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    else if (x <= 650.0d0) then
        tmp = ((2.0d0 + (x * 2.0d0)) * t_0) / 2.0d0
    else if (x <= 5d+102) then
        tmp = 0.0d0
    else if (x <= 1d+257) then
        tmp = x * t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (x * ((x * 0.5) + -1.0));
	double tmp;
	if (x <= -700000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 650.0) {
		tmp = ((2.0 + (x * 2.0)) * t_0) / 2.0;
	} else if (x <= 5e+102) {
		tmp = 0.0;
	} else if (x <= 1e+257) {
		tmp = x * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + (x * ((x * 0.5) + -1.0))
	tmp = 0
	if x <= -700000000.0:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	elif x <= 650.0:
		tmp = ((2.0 + (x * 2.0)) * t_0) / 2.0
	elif x <= 5e+102:
		tmp = 0.0
	elif x <= 1e+257:
		tmp = x * t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0)))
	tmp = 0.0
	if (x <= -700000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	elseif (x <= 650.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(x * 2.0)) * t_0) / 2.0);
	elseif (x <= 5e+102)
		tmp = 0.0;
	elseif (x <= 1e+257)
		tmp = Float64(x * t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 + (x * ((x * 0.5) + -1.0));
	tmp = 0.0;
	if (x <= -700000000.0)
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	elseif (x <= 650.0)
		tmp = ((2.0 + (x * 2.0)) * t_0) / 2.0;
	elseif (x <= 5e+102)
		tmp = 0.0;
	elseif (x <= 1e+257)
		tmp = x * t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -700000000.0], 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, 650.0], N[(N[(N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+102], 0.0, If[LessEqual[x, 1e+257], N[(x * t$95$0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7e8

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. pow1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

   10. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   12. Simplified 1.6%

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

   13. Taylor expanded in x around 0 75.8%

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

       1. *-commutative 75.8%

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

   15. Simplified 75.8%

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

   if -7e8 < x < 650

   1. Initial program 51.9%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in eps around 0 26.9%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. associate-*r* 75.6%

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

      6. distribute-rgt-out 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in x around 0 75.3%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} \cdot \left(2 + 2 \cdot x\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in eps around 0 75.3%

      \[\leadsto \frac{\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right) \cdot \left(2 + 2 \cdot x\right)}}{2} \]

   if 650 < x < 5e102 or 1.00000000000000003e257 < 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 70.5%

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

      1. div-sub 70.5%

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

      2. mul-1-neg 70.5%

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

      3. rec-exp 70.5%

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

      4. +-inverses 70.5%

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

      5. metadata-eval 70.5%

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

   7. Simplified 70.5%

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

   if 5e102 < x < 1.00000000000000003e257

   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 35.4%

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

      1. associate-+r+ 35.4%

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

      2. mul-1-neg 35.4%

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

      3. sub-neg 35.4%

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

      4. +-inverses 35.4%

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

      5. associate-*r* 35.4%

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

      6. distribute-rgt-out 35.4%

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

      7. mul-1-neg 35.4%

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

   7. Simplified 35.4%

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

   8. Taylor expanded in x around inf 35.4%

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

   9. Taylor expanded in x around 0 66.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -700000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;\frac{\left(2 + x \cdot 2\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+257}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+111}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -75000000.0)
   (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))
   (if (<= x 490.0)
     1.0
     (if (<= x 1.12e+111)
       0.0
       (if (<= x 2e+255) (* x (+ 1.0 (* x (+ (* x 0.5) -1.0)))) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 490.0) {
		tmp = 1.0;
	} else if (x <= 1.12e+111) {
		tmp = 0.0;
	} else if (x <= 2e+255) {
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-75000000.0d0)) then
        tmp = x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0))))))
    else if (x <= 490.0d0) then
        tmp = 1.0d0
    else if (x <= 1.12d+111) then
        tmp = 0.0d0
    else if (x <= 2d+255) then
        tmp = x * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -75000000.0) {
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	} else if (x <= 490.0) {
		tmp = 1.0;
	} else if (x <= 1.12e+111) {
		tmp = 0.0;
	} else if (x <= 2e+255) {
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -75000000.0:
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))
	elif x <= 490.0:
		tmp = 1.0
	elif x <= 1.12e+111:
		tmp = 0.0
	elif x <= 2e+255:
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -75000000.0)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))));
	elseif (x <= 490.0)
		tmp = 1.0;
	elseif (x <= 1.12e+111)
		tmp = 0.0;
	elseif (x <= 2e+255)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -75000000.0)
		tmp = x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))));
	elseif (x <= 490.0)
		tmp = 1.0;
	elseif (x <= 1.12e+111)
		tmp = 0.0;
	elseif (x <= 2e+255)
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -75000000.0], 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, 490.0], 1.0, If[LessEqual[x, 1.12e+111], 0.0, If[LessEqual[x, 2e+255], N[(x * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.5e7

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. pow1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

   10. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   12. Simplified 1.6%

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

   13. Taylor expanded in x around 0 75.8%

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

       1. *-commutative 75.8%

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

   15. Simplified 75.8%

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

   if -7.5e7 < x < 490

   1. Initial program 51.9%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in eps around 0 26.9%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. associate-*r* 75.6%

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

      6. distribute-rgt-out 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in x around 0 75.1%

      \[\leadsto \frac{\frac{0 + \color{blue}{2 \cdot \varepsilon}}{\varepsilon}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 75.1%

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

   10. Simplified 75.1%

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

   11. Taylor expanded in eps around 0 75.1%

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

   if 490 < x < 1.11999999999999995e111 or 1.99999999999999998e255 < 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 70.5%

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

      1. div-sub 70.5%

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

      2. mul-1-neg 70.5%

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

      3. rec-exp 70.5%

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

      4. +-inverses 70.5%

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

      5. metadata-eval 70.5%

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

   7. Simplified 70.5%

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

   if 1.11999999999999995e111 < x < 1.99999999999999998e255

   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 35.4%

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

      1. associate-+r+ 35.4%

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

      2. mul-1-neg 35.4%

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

      3. sub-neg 35.4%

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

      4. +-inverses 35.4%

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

      5. associate-*r* 35.4%

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

      6. distribute-rgt-out 35.4%

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

      7. mul-1-neg 35.4%

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

   7. Simplified 35.4%

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

   8. Taylor expanded in x around inf 35.4%

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

   9. Taylor expanded in x around 0 66.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+111}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+255}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+175}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+101}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+258}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.15e+175)
   (* x (+ x 1.0))
   (if (<= x 1.0)
     (/ (- 2.0 (* x (+ eps 2.0))) 2.0)
     (if (<= x 4e+101)
       0.0
       (if (<= x 1.46e+258) (* x (+ 1.0 (* x (+ (* x 0.5) -1.0)))) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.15e+175) {
		tmp = x * (x + 1.0);
	} else if (x <= 1.0) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else if (x <= 4e+101) {
		tmp = 0.0;
	} else if (x <= 1.46e+258) {
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.15d+175)) then
        tmp = x * (x + 1.0d0)
    else if (x <= 1.0d0) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 2.0d0
    else if (x <= 4d+101) then
        tmp = 0.0d0
    else if (x <= 1.46d+258) then
        tmp = x * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.15e+175) {
		tmp = x * (x + 1.0);
	} else if (x <= 1.0) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else if (x <= 4e+101) {
		tmp = 0.0;
	} else if (x <= 1.46e+258) {
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.15e+175:
		tmp = x * (x + 1.0)
	elif x <= 1.0:
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0
	elif x <= 4e+101:
		tmp = 0.0
	elif x <= 1.46e+258:
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.15e+175)
		tmp = Float64(x * Float64(x + 1.0));
	elseif (x <= 1.0)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 2.0))) / 2.0);
	elseif (x <= 4e+101)
		tmp = 0.0;
	elseif (x <= 1.46e+258)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.15e+175)
		tmp = x * (x + 1.0);
	elseif (x <= 1.0)
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	elseif (x <= 4e+101)
		tmp = 0.0;
	elseif (x <= 1.46e+258)
		tmp = x * (1.0 + (x * ((x * 0.5) + -1.0)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.15e+175], N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(N[(2.0 - N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+101], 0.0, If[LessEqual[x, 1.46e+258], N[(x * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.15e175

   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. associate-*r* 0.0%

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

      6. distribute-rgt-out 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in x around inf 0.0%

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

      1. pow1 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 0.0%

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

      4. sqr-neg 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 1.6%

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

   10. Applied egg-rr 1.6%

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

       1. unpow1 1.6%

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

   12. Simplified 1.6%

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

   13. Taylor expanded in x around 0 100.0%

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

       1. +-commutative 100.0%

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

   15. Simplified 100.0%

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

   if -1.15e175 < x < 1

   1. Initial program 56.4%

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

   3. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 79.7%

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

   7. Taylor expanded in eps around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. unsub-neg 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in x around 0 68.7%

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

       1. mul-1-neg 68.7%

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

       2. unsub-neg 68.7%

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

   12. Simplified 68.7%

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

   if 1 < x < 3.9999999999999999e101 or 1.46e258 < 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 68.2%

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

      1. div-sub 68.2%

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

      2. mul-1-neg 68.2%

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

      3. rec-exp 68.2%

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

      4. +-inverses 68.2%

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

      5. metadata-eval 68.2%

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

   7. Simplified 68.2%

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

   if 3.9999999999999999e101 < x < 1.46e258

   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 35.4%

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

      1. associate-+r+ 35.4%

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

      2. mul-1-neg 35.4%

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

      3. sub-neg 35.4%

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

      4. +-inverses 35.4%

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

      5. associate-*r* 35.4%

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

      6. distribute-rgt-out 35.4%

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

      7. mul-1-neg 35.4%

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

   7. Simplified 35.4%

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

   8. Taylor expanded in x around inf 35.4%

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

   9. Taylor expanded in x around 0 66.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+175}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+101}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+258}:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+175}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+144}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ x 1.0))))
   (if (<= x -1.15e+175)
     t_0
     (if (<= x 1.0)
       (/ (- 2.0 (* x (+ eps 2.0))) 2.0)
       (if (<= x 1.7e+144) 0.0 (if (<= x 5e+258) t_0 0.0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -1.15e+175) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else if (x <= 1.7e+144) {
		tmp = 0.0;
	} else if (x <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x + 1.0d0)
    if (x <= (-1.15d+175)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 2.0d0
    else if (x <= 1.7d+144) then
        tmp = 0.0d0
    else if (x <= 5d+258) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -1.15e+175) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else if (x <= 1.7e+144) {
		tmp = 0.0;
	} else if (x <= 5e+258) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x + 1.0)
	tmp = 0
	if x <= -1.15e+175:
		tmp = t_0
	elif x <= 1.0:
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0
	elif x <= 1.7e+144:
		tmp = 0.0
	elif x <= 5e+258:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.15e+175)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 2.0))) / 2.0);
	elseif (x <= 1.7e+144)
		tmp = 0.0;
	elseif (x <= 5e+258)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x + 1.0);
	tmp = 0.0;
	if (x <= -1.15e+175)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	elseif (x <= 1.7e+144)
		tmp = 0.0;
	elseif (x <= 5e+258)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e+175], t$95$0, If[LessEqual[x, 1.0], N[(N[(2.0 - N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.7e+144], 0.0, If[LessEqual[x, 5e+258], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15e175 or 1.7e144 < x < 5e258

   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 18.6%

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

      1. associate-+r+ 18.6%

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

      2. mul-1-neg 18.6%

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

      3. sub-neg 18.6%

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

      4. +-inverses 18.6%

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

      5. associate-*r* 18.6%

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

      6. distribute-rgt-out 18.6%

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

      7. mul-1-neg 18.6%

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

   7. Simplified 18.6%

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

   8. Taylor expanded in x around inf 18.6%

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

      1. pow1 18.6%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 38.3%

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

      4. sqr-neg 38.3%

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

      5. sqrt-unprod 38.3%

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

      6. add-sqr-sqrt 39.0%

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

   10. Applied egg-rr 39.0%

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

       1. unpow1 39.0%

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

   12. Simplified 39.0%

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

   13. Taylor expanded in x around 0 80.5%

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

       1. +-commutative 80.5%

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

   15. Simplified 80.5%

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

   if -1.15e175 < x < 1

   1. Initial program 56.4%

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

   3. Simplified 44.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 79.7%

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

   7. Taylor expanded in eps around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. unsub-neg 79.9%

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

   9. Simplified 79.9%

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

   10. Taylor expanded in x around 0 68.7%

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

       1. mul-1-neg 68.7%

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

       2. unsub-neg 68.7%

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

   12. Simplified 68.7%

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

   if 1 < x < 1.7e144 or 5e258 < 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 66.2%

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

      1. div-sub 66.2%

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

      2. mul-1-neg 66.2%

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

      3. rec-exp 66.3%

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

      4. +-inverses 66.3%

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

      5. metadata-eval 66.3%

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

   7. Simplified 66.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+175}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+144}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+258}:\\ \;\;\;\;x \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot 2}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+144}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+258}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ x 1.0))))
   (if (<= x -6.5e+128)
     t_0
     (if (<= x 550.0)
       (/ (/ (* eps 2.0) eps) 2.0)
       (if (<= x 1.7e+144) 0.0 (if (<= x 1.5e+258) t_0 0.0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -6.5e+128) {
		tmp = t_0;
	} else if (x <= 550.0) {
		tmp = ((eps * 2.0) / eps) / 2.0;
	} else if (x <= 1.7e+144) {
		tmp = 0.0;
	} else if (x <= 1.5e+258) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x + 1.0d0)
    if (x <= (-6.5d+128)) then
        tmp = t_0
    else if (x <= 550.0d0) then
        tmp = ((eps * 2.0d0) / eps) / 2.0d0
    else if (x <= 1.7d+144) then
        tmp = 0.0d0
    else if (x <= 1.5d+258) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -6.5e+128) {
		tmp = t_0;
	} else if (x <= 550.0) {
		tmp = ((eps * 2.0) / eps) / 2.0;
	} else if (x <= 1.7e+144) {
		tmp = 0.0;
	} else if (x <= 1.5e+258) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x + 1.0)
	tmp = 0
	if x <= -6.5e+128:
		tmp = t_0
	elif x <= 550.0:
		tmp = ((eps * 2.0) / eps) / 2.0
	elif x <= 1.7e+144:
		tmp = 0.0
	elif x <= 1.5e+258:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x + 1.0))
	tmp = 0.0
	if (x <= -6.5e+128)
		tmp = t_0;
	elseif (x <= 550.0)
		tmp = Float64(Float64(Float64(eps * 2.0) / eps) / 2.0);
	elseif (x <= 1.7e+144)
		tmp = 0.0;
	elseif (x <= 1.5e+258)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x + 1.0);
	tmp = 0.0;
	if (x <= -6.5e+128)
		tmp = t_0;
	elseif (x <= 550.0)
		tmp = ((eps * 2.0) / eps) / 2.0;
	elseif (x <= 1.7e+144)
		tmp = 0.0;
	elseif (x <= 1.5e+258)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+128], t$95$0, If[LessEqual[x, 550.0], N[(N[(N[(eps * 2.0), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.7e+144], 0.0, If[LessEqual[x, 1.5e+258], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000003e128 or 1.7e144 < x < 1.5e258

   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 17.9%

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

      1. associate-+r+ 17.9%

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

      2. mul-1-neg 17.9%

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

      3. sub-neg 17.9%

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

      4. +-inverses 17.9%

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

      5. associate-*r* 17.9%

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

      6. distribute-rgt-out 17.9%

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

      7. mul-1-neg 17.9%

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

   7. Simplified 17.9%

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

   8. Taylor expanded in x around inf 17.9%

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

      1. pow1 17.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 36.8%

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

      4. sqr-neg 36.8%

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

      5. sqrt-unprod 36.8%

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

      6. add-sqr-sqrt 37.5%

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

   10. Applied egg-rr 37.5%

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

       1. unpow1 37.5%

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

   12. Simplified 37.5%

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

   13. Taylor expanded in x around 0 81.2%

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

       1. +-commutative 81.2%

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

   15. Simplified 81.2%

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

   if -6.5000000000000003e128 < x < 550

   1. Initial program 56.1%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in eps around 0 24.5%

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

      1. associate-+r+ 68.9%

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

      2. mul-1-neg 68.9%

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

      3. sub-neg 68.9%

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

      4. +-inverses 68.9%

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

      5. associate-*r* 68.9%

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

      6. distribute-rgt-out 69.5%

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

      7. mul-1-neg 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in x around 0 69.3%

      \[\leadsto \frac{\frac{0 + \color{blue}{2 \cdot \varepsilon}}{\varepsilon}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 69.3%

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

   10. Simplified 69.3%

       \[\leadsto \frac{\frac{0 + \color{blue}{\varepsilon \cdot 2}}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. +-lft-identity 69.3%

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

   12. Applied egg-rr 69.3%

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

   if 550 < x < 1.7e144 or 1.5e258 < 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 68.2%

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

      1. div-sub 68.2%

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

      2. mul-1-neg 68.2%

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

      3. rec-exp 68.2%

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

      4. +-inverses 68.2%

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

      5. metadata-eval 68.2%

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

   7. Simplified 68.2%

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

Alternative 14: 63.8% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x + 1\right)\\ \mathbf{if}\;x \leq -75000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 510:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+144}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+260}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ x 1.0))))
   (if (<= x -75000000.0)
     t_0
     (if (<= x 510.0)
       1.0
       (if (<= x 1.7e+144) 0.0 (if (<= x 5e+260) t_0 0.0))))))

C

double code(double x, double eps) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -75000000.0) {
		tmp = t_0;
	} else if (x <= 510.0) {
		tmp = 1.0;
	} else if (x <= 1.7e+144) {
		tmp = 0.0;
	} else if (x <= 5e+260) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x + 1.0d0)
    if (x <= (-75000000.0d0)) then
        tmp = t_0
    else if (x <= 510.0d0) then
        tmp = 1.0d0
    else if (x <= 1.7d+144) then
        tmp = 0.0d0
    else if (x <= 5d+260) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (x + 1.0);
	double tmp;
	if (x <= -75000000.0) {
		tmp = t_0;
	} else if (x <= 510.0) {
		tmp = 1.0;
	} else if (x <= 1.7e+144) {
		tmp = 0.0;
	} else if (x <= 5e+260) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (x + 1.0)
	tmp = 0
	if x <= -75000000.0:
		tmp = t_0
	elif x <= 510.0:
		tmp = 1.0
	elif x <= 1.7e+144:
		tmp = 0.0
	elif x <= 5e+260:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(x + 1.0))
	tmp = 0.0
	if (x <= -75000000.0)
		tmp = t_0;
	elseif (x <= 510.0)
		tmp = 1.0;
	elseif (x <= 1.7e+144)
		tmp = 0.0;
	elseif (x <= 5e+260)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (x + 1.0);
	tmp = 0.0;
	if (x <= -75000000.0)
		tmp = t_0;
	elseif (x <= 510.0)
		tmp = 1.0;
	elseif (x <= 1.7e+144)
		tmp = 0.0;
	elseif (x <= 5e+260)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -75000000.0], t$95$0, If[LessEqual[x, 510.0], 1.0, If[LessEqual[x, 1.7e+144], 0.0, If[LessEqual[x, 5e+260], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e7 or 1.7e144 < x < 4.9999999999999996e260

   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 13.9%

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

      1. associate-+r+ 13.9%

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

      2. mul-1-neg 13.9%

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

      3. sub-neg 13.9%

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

      4. +-inverses 13.9%

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

      5. associate-*r* 13.9%

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

      6. distribute-rgt-out 13.9%

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

      7. mul-1-neg 13.9%

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

   7. Simplified 13.9%

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

   8. Taylor expanded in x around inf 13.9%

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

      1. pow1 13.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 28.6%

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

      4. sqr-neg 28.6%

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

      5. sqrt-unprod 28.6%

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

      6. add-sqr-sqrt 29.5%

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

   10. Applied egg-rr 29.5%

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

       1. unpow1 29.5%

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

   12. Simplified 29.5%

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

   13. Taylor expanded in x around 0 64.1%

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

       1. +-commutative 64.1%

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

   15. Simplified 64.1%

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

   if -7.5e7 < x < 510

   1. Initial program 51.9%

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

   3. Simplified 30.3%

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

   5. Taylor expanded in eps around 0 26.9%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. associate-*r* 75.6%

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

      6. distribute-rgt-out 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in x around 0 75.1%

      \[\leadsto \frac{\frac{0 + \color{blue}{2 \cdot \varepsilon}}{\varepsilon}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 75.1%

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

   10. Simplified 75.1%

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

   11. Taylor expanded in eps around 0 75.1%

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

   if 510 < x < 1.7e144 or 4.9999999999999996e260 < 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 68.2%

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

      1. div-sub 68.2%

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

      2. mul-1-neg 68.2%

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

      3. rec-exp 68.2%

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

      4. +-inverses 68.2%

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

      5. metadata-eval 68.2%

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

   7. Simplified 68.2%

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

Alternative 15: 56.9% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 610:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 610.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 610.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 610.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 610.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 610.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 610.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 610.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 610.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 610

   1. Initial program 61.5%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in eps around 0 21.5%

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

      1. associate-+r+ 60.4%

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

      2. mul-1-neg 60.4%

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

      3. sub-neg 60.4%

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

      4. +-inverses 60.4%

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

      5. associate-*r* 60.4%

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

      6. distribute-rgt-out 60.9%

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

      7. mul-1-neg 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in x around 0 61.1%

      \[\leadsto \frac{\frac{0 + \color{blue}{2 \cdot \varepsilon}}{\varepsilon}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 61.1%

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

   10. Simplified 61.1%

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

   11. Taylor expanded in eps around 0 60.6%

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

   if 610 < 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 52.4%

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

      1. div-sub 52.4%

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

      2. mul-1-neg 52.4%

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

      3. rec-exp 52.4%

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

      4. +-inverses 52.4%

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

      5. metadata-eval 52.4%

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

   7. Simplified 52.4%

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

Alternative 16: 15.8% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 70.9%

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

3. Simplified 62.7%

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

5. Taylor expanded in eps around 0 14.3%

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

   1. div-sub 14.3%

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

   2. mul-1-neg 14.3%

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

   3. rec-exp 14.3%

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

   4. +-inverses 14.6%

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

   5. metadata-eval 14.6%

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

7. Simplified 14.6%

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

Reproduce

herbie shell --seed 2024145 
(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))