NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 98.9%

Time: 12.6s

Alternatives: 16

Speedup: 1.1×

• 38.2% 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: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 67.1%

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

4. Taylor expanded in eps around inf 99.2%

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

5. Final simplification 99.2%

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

Alternative 2: 85.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-37}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1e-37)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ (exp (* x (+ eps -1.0))) (exp (* eps (- x)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-37) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 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 (eps <= 1d-37) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1e-37) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1e-37:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1e-37)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1e-37)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1e-37], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.00000000000000007e-37

   1. Initial program 63.9%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in eps around inf 98.8%

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

   5. Taylor expanded in eps around 0 75.0%

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

      1. rec-exp 75.0%

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

      2. neg-mul-1 75.0%

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

      3. count-2 75.0%

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

      4. neg-mul-1 75.0%

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

   7. Simplified 75.0%

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

   if 1.00000000000000007e-37 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. associate-*r* 100.0%

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

      7. neg-mul-1 100.0%

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

      8. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-37}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (exp((x * (eps + -1.0))) + exp((x * (-1.0 - 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)))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in eps around inf 99.2%

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

5. Taylor expanded in eps around -inf 99.2%

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

   1. associate-*r* 99.2%

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

   2. neg-mul-1 99.2%

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

   3. mul-1-neg 99.2%

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

   4. sub-neg 99.2%

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

   5. mul-1-neg 99.2%

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

   6. associate-*r* 99.2%

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

   7. neg-mul-1 99.2%

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

   8. mul-1-neg 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 58.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(1 - \varepsilon\right)} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.5e-134)
   (/
    (+ (* (+ 1.0 (/ 1.0 eps)) (exp (* x (- 1.0 eps)))) (+ 1.0 (/ -1.0 eps)))
    2.0)
   (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.5e-134) {
		tmp = (((1.0 + (1.0 / eps)) * exp((x * (1.0 - eps)))) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-7.5d-134)) then
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp((x * (1.0d0 - eps)))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7.5e-134) {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp((x * (1.0 - eps)))) + (1.0 + (-1.0 / eps))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7.5e-134:
		tmp = (((1.0 + (1.0 / eps)) * math.exp((x * (1.0 - eps)))) + (1.0 + (-1.0 / eps))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.5e-134)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(1.0 - eps)))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7.5e-134)
		tmp = (((1.0 + (1.0 / eps)) * exp((x * (1.0 - eps)))) + (1.0 + (-1.0 / eps))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.5e-134], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.50000000000000048e-134

   1. Initial program 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in x around 0 54.7%

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

      1. add-log-exp 54.7%

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

      2. *-un-lft-identity 54.7%

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

      3. log-prod 54.7%

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

      4. metadata-eval 54.7%

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

      5. add-log-exp 54.7%

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

      6. *-commutative 54.7%

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

      7. add-sqr-sqrt 54.7%

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

      8. sqrt-unprod 54.7%

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

      9. sqr-neg 54.7%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 37.5%

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

   6. Applied egg-rr 37.5%

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

      1. +-lft-identity 37.5%

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

   8. Simplified 37.5%

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

   if -7.50000000000000048e-134 < x

   1. Initial program 72.5%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in x around 0 66.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(1 - \varepsilon\right)} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 5: 59.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.4e-132)
   (/
    (+ (+ 1.0 (/ 1.0 eps)) (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
    2.0)
   (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.4e-132) {
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-4.4d-132)) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.4e-132) {
		tmp = ((1.0 + (1.0 / eps)) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.4e-132:
		tmp = ((1.0 + (1.0 / eps)) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.4e-132)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.4e-132)
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.4e-132], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.39999999999999981e-132

   1. Initial program 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in x around 0 39.0%

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

   if -4.39999999999999981e-132 < x

   1. Initial program 72.5%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in x around 0 66.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 6: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-290}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.5e-290)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-290) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-1.5d-290)) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.5e-290) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.5e-290:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.5e-290)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.5e-290)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.5e-290], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.49999999999999996e-290

   1. Initial program 72.8%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in eps around inf 98.7%

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

   5. Taylor expanded in eps around 0 77.4%

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

      1. rec-exp 77.4%

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

      2. neg-mul-1 77.4%

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

      3. count-2 77.4%

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

      4. neg-mul-1 77.4%

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

   7. Simplified 77.4%

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

   if -1.49999999999999996e-290 < x

   1. Initial program 76.4%

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

   3. Simplified 70.3%

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

   4. Taylor expanded in eps around inf 99.4%

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

   5. Taylor expanded in x around 0 60.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-290}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 7: 71.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-6) (/ (* 2.0 (exp (- x))) 2.0) (/ (+ 1.0 (exp (* x eps))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * eps))) / 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 <= (-9d-6)) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-6:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((x * eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-6], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in eps around inf 97.1%

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

   5. Taylor expanded in eps around 0 94.2%

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

      1. rec-exp 94.2%

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

      2. neg-mul-1 94.2%

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

      3. count-2 94.2%

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

      4. neg-mul-1 94.2%

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

   7. Simplified 94.2%

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

   if -9.00000000000000023e-6 < x

   1. Initial program 71.6%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in x around 0 67.6%

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

   6. Taylor expanded in eps around inf 67.7%

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

      1. *-commutative 67.7%

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

   8. Simplified 67.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]

Alternative 8: 70.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.9 \cdot 10^{+263}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{+304}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.9e+263)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (<= eps 4.2e+304)
     (/ (+ 2.0 (* x (* (- 1.0 eps) (+ -1.0 (/ -1.0 eps))))) 2.0)
     (* x (* eps -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.9e+263) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (eps <= 4.2e+304) {
		tmp = (2.0 + (x * ((1.0 - eps) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else {
		tmp = x * (eps * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.9d+263) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (eps <= 4.2d+304) then
        tmp = (2.0d0 + (x * ((1.0d0 - eps) * ((-1.0d0) + ((-1.0d0) / eps))))) / 2.0d0
    else
        tmp = x * (eps * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.9e+263) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (eps <= 4.2e+304) {
		tmp = (2.0 + (x * ((1.0 - eps) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else {
		tmp = x * (eps * -0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.9e+263:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif eps <= 4.2e+304:
		tmp = (2.0 + (x * ((1.0 - eps) * (-1.0 + (-1.0 / eps))))) / 2.0
	else:
		tmp = x * (eps * -0.5)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.9e+263)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (eps <= 4.2e+304)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - eps) * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	else
		tmp = Float64(x * Float64(eps * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.9e+263)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (eps <= 4.2e+304)
		tmp = (2.0 + (x * ((1.0 - eps) * (-1.0 + (-1.0 / eps))))) / 2.0;
	else
		tmp = x * (eps * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.9e+263], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 4.2e+304], N[(N[(2.0 + N[(x * N[(N[(1.0 - eps), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.9e263

   1. Initial program 73.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in eps around inf 99.1%

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

   5. Taylor expanded in eps around 0 70.6%

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

      1. rec-exp 70.5%

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

      2. neg-mul-1 70.5%

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

      3. count-2 70.5%

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

      4. neg-mul-1 70.5%

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

   7. Simplified 70.5%

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

   if 1.9e263 < eps < 4.1999999999999999e304

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 51.6%

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

   5. Taylor expanded in x around 0 27.0%

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

      1. mul-1-neg 27.0%

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

      2. distribute-rgt-neg-in 27.0%

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

      3. *-commutative 27.0%

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

      4. distribute-rgt-neg-in 27.0%

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

      5. mul-1-neg 27.0%

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

      6. distribute-lft-in 27.0%

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

      7. metadata-eval 27.0%

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

      8. neg-mul-1 27.0%

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

      9. distribute-neg-frac 27.0%

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

      10. metadata-eval 27.0%

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

   7. Simplified 27.0%

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

   if 4.1999999999999999e304 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.9 \cdot 10^{+263}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{+304}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \end{array} \]

Alternative 9: 59.7% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon + 1\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + -1\right) \cdot t_0}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+229}:\\ \;\;\;\;\frac{t_0 + \frac{t_0}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ eps 1.0))))
   (if (<= x -9e-6)
     (/ (* (+ (/ 1.0 eps) -1.0) t_0) 2.0)
     (if (<= x 520.0)
       1.0
       (if (<= x 4.1e+195)
         0.0
         (if (<= x 5.5e+229)
           (/ (+ t_0 (/ t_0 eps)) 2.0)
           (/ (+ (+ 1.0 (/ 1.0 eps)) (+ 1.0 (/ -1.0 eps))) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (eps + 1.0);
	double tmp;
	if (x <= -9e-6) {
		tmp = (((1.0 / eps) + -1.0) * t_0) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 4.1e+195) {
		tmp = 0.0;
	} else if (x <= 5.5e+229) {
		tmp = (t_0 + (t_0 / eps)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	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 * (eps + 1.0d0)
    if (x <= (-9d-6)) then
        tmp = (((1.0d0 / eps) + (-1.0d0)) * t_0) / 2.0d0
    else if (x <= 520.0d0) then
        tmp = 1.0d0
    else if (x <= 4.1d+195) then
        tmp = 0.0d0
    else if (x <= 5.5d+229) then
        tmp = (t_0 + (t_0 / eps)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (eps + 1.0);
	double tmp;
	if (x <= -9e-6) {
		tmp = (((1.0 / eps) + -1.0) * t_0) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 4.1e+195) {
		tmp = 0.0;
	} else if (x <= 5.5e+229) {
		tmp = (t_0 + (t_0 / eps)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (eps + 1.0)
	tmp = 0
	if x <= -9e-6:
		tmp = (((1.0 / eps) + -1.0) * t_0) / 2.0
	elif x <= 520.0:
		tmp = 1.0
	elif x <= 4.1e+195:
		tmp = 0.0
	elif x <= 5.5e+229:
		tmp = (t_0 + (t_0 / eps)) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps + 1.0))
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + -1.0) * t_0) / 2.0);
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 4.1e+195)
		tmp = 0.0;
	elseif (x <= 5.5e+229)
		tmp = Float64(Float64(t_0 + Float64(t_0 / eps)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (eps + 1.0);
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = (((1.0 / eps) + -1.0) * t_0) / 2.0;
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 4.1e+195)
		tmp = 0.0;
	elseif (x <= 5.5e+229)
		tmp = (t_0 + (t_0 / eps)) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e-6], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, If[LessEqual[x, 4.1e+195], 0.0, If[LessEqual[x, 5.5e+229], N[(N[(t$95$0 + N[(t$95$0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in x around 0 53.6%

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

   5. Taylor expanded in x around inf 24.3%

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

      1. sub-neg 24.3%

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

      2. metadata-eval 24.3%

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

      3. associate-*r* 24.3%

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

      4. +-commutative 24.3%

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

   7. Simplified 24.3%

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

   if -9.00000000000000023e-6 < x < 520

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around 0 71.6%

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

   if 520 < x < 4.1e195

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 49.7%

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

   8. Simplified 49.7%

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

   if 4.1e195 < x < 5.5000000000000002e229

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 83.3%

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

   5. Taylor expanded in x around inf 50.0%

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

      1. sub-neg 50.0%

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

      2. metadata-eval 50.0%

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

      3. associate-*r* 50.0%

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

      4. +-commutative 50.0%

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

   7. Simplified 50.0%

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

      1. +-commutative 50.0%

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

      2. distribute-rgt-in 50.0%

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

      3. associate-*l/ 50.0%

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

      4. *-un-lft-identity 50.0%

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

      5. associate-*r* 50.0%

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

      6. neg-mul-1 50.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 50.0%

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

      9. sqr-neg 50.0%

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

      10. sqrt-unprod 19.4%

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

      11. add-sqr-sqrt 19.4%

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

   9. Applied egg-rr 19.4%

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

   if 5.5000000000000002e229 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 28.4%

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

   5. Taylor expanded in x around 0 54.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + -1\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+229}:\\ \;\;\;\;\frac{x \cdot \left(\varepsilon + 1\right) + \frac{x \cdot \left(\varepsilon + 1\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 10: 59.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + -1\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+230}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-6)
   (/ (* (+ (/ 1.0 eps) -1.0) (* x (+ eps 1.0))) 2.0)
   (if (<= x 620.0)
     1.0
     (if (<= x 1.3e+195)
       0.0
       (if (<= x 2.6e+230)
         (* (* x eps) 0.5)
         (/ (+ (+ 1.0 (/ 1.0 eps)) (+ 1.0 (/ -1.0 eps))) 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0;
	} else if (x <= 620.0) {
		tmp = 1.0;
	} else if (x <= 1.3e+195) {
		tmp = 0.0;
	} else if (x <= 2.6e+230) {
		tmp = (x * eps) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 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 <= (-9d-6)) then
        tmp = (((1.0d0 / eps) + (-1.0d0)) * (x * (eps + 1.0d0))) / 2.0d0
    else if (x <= 620.0d0) then
        tmp = 1.0d0
    else if (x <= 1.3d+195) then
        tmp = 0.0d0
    else if (x <= 2.6d+230) then
        tmp = (x * eps) * 0.5d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0;
	} else if (x <= 620.0) {
		tmp = 1.0;
	} else if (x <= 1.3e+195) {
		tmp = 0.0;
	} else if (x <= 2.6e+230) {
		tmp = (x * eps) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-6:
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0
	elif x <= 620.0:
		tmp = 1.0
	elif x <= 1.3e+195:
		tmp = 0.0
	elif x <= 2.6e+230:
		tmp = (x * eps) * 0.5
	else:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + -1.0) * Float64(x * Float64(eps + 1.0))) / 2.0);
	elseif (x <= 620.0)
		tmp = 1.0;
	elseif (x <= 1.3e+195)
		tmp = 0.0;
	elseif (x <= 2.6e+230)
		tmp = Float64(Float64(x * eps) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0;
	elseif (x <= 620.0)
		tmp = 1.0;
	elseif (x <= 1.3e+195)
		tmp = 0.0;
	elseif (x <= 2.6e+230)
		tmp = (x * eps) * 0.5;
	else
		tmp = ((1.0 + (1.0 / eps)) + (1.0 + (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-6], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 620.0], 1.0, If[LessEqual[x, 1.3e+195], 0.0, If[LessEqual[x, 2.6e+230], N[(N[(x * eps), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in x around 0 53.6%

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

   5. Taylor expanded in x around inf 24.3%

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

      1. sub-neg 24.3%

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

      2. metadata-eval 24.3%

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

      3. associate-*r* 24.3%

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

      4. +-commutative 24.3%

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

   7. Simplified 24.3%

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

   if -9.00000000000000023e-6 < x < 620

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around 0 71.6%

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

   if 620 < x < 1.30000000000000001e195

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 49.7%

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

   8. Simplified 49.7%

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

   if 1.30000000000000001e195 < x < 2.5999999999999999e230

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 83.3%

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

   5. Taylor expanded in eps around inf 50.0%

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

      1. associate-*r* 50.0%

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

      2. mul-1-neg 50.0%

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

   7. Simplified 50.0%

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

      1. div-inv 50.0%

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

      2. *-commutative 50.0%

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

      3. add-sqr-sqrt 50.0%

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

      4. sqrt-unprod 69.4%

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

      5. sqr-neg 69.4%

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

      6. sqrt-unprod 19.4%

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

      7. add-sqr-sqrt 19.4%

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

      8. metadata-eval 19.4%

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

   9. Applied egg-rr 19.4%

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

   if 2.5999999999999999e230 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 28.4%

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

   5. Taylor expanded in x around 0 54.8%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + -1\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+230}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 11: 57.7% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + -1\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.02 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-6)
   (/ (* (+ (/ 1.0 eps) -1.0) (* x (+ eps 1.0))) 2.0)
   (if (<= x 550.0)
     1.0
     (if (<= x 2.02e+195) 0.0 (if (<= x 1.3e+297) (* (* x eps) 0.5) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 2.02e+195) {
		tmp = 0.0;
	} else if (x <= 1.3e+297) {
		tmp = (x * eps) * 0.5;
	} 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 <= (-9d-6)) then
        tmp = (((1.0d0 / eps) + (-1.0d0)) * (x * (eps + 1.0d0))) / 2.0d0
    else if (x <= 550.0d0) then
        tmp = 1.0d0
    else if (x <= 2.02d+195) then
        tmp = 0.0d0
    else if (x <= 1.3d+297) then
        tmp = (x * eps) * 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 2.02e+195) {
		tmp = 0.0;
	} else if (x <= 1.3e+297) {
		tmp = (x * eps) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-6:
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0
	elif x <= 550.0:
		tmp = 1.0
	elif x <= 2.02e+195:
		tmp = 0.0
	elif x <= 1.3e+297:
		tmp = (x * eps) * 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + -1.0) * Float64(x * Float64(eps + 1.0))) / 2.0);
	elseif (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 2.02e+195)
		tmp = 0.0;
	elseif (x <= 1.3e+297)
		tmp = Float64(Float64(x * eps) * 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = (((1.0 / eps) + -1.0) * (x * (eps + 1.0))) / 2.0;
	elseif (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 2.02e+195)
		tmp = 0.0;
	elseif (x <= 1.3e+297)
		tmp = (x * eps) * 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-6], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, If[LessEqual[x, 2.02e+195], 0.0, If[LessEqual[x, 1.3e+297], N[(N[(x * eps), $MachinePrecision] * 0.5), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in x around 0 53.6%

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

   5. Taylor expanded in x around inf 24.3%

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

      1. sub-neg 24.3%

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

      2. metadata-eval 24.3%

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

      3. associate-*r* 24.3%

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

      4. +-commutative 24.3%

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

   7. Simplified 24.3%

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

   if -9.00000000000000023e-6 < x < 550

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around 0 71.6%

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

   if 550 < x < 2.01999999999999988e195 or 1.3e297 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 50.8%

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

   8. Simplified 50.8%

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

   if 2.01999999999999988e195 < x < 1.3e297

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 40.5%

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

   5. Taylor expanded in eps around inf 30.9%

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

      1. associate-*r* 30.9%

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

      2. mul-1-neg 30.9%

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

   7. Simplified 30.9%

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

      1. div-inv 30.9%

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

      2. *-commutative 30.9%

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

      3. add-sqr-sqrt 30.5%

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

      4. sqrt-unprod 71.2%

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

      5. sqr-neg 71.2%

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

      6. sqrt-unprod 26.2%

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

      7. add-sqr-sqrt 26.7%

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

      8. metadata-eval 26.7%

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

   9. Applied egg-rr 26.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + -1\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.02 \cdot 10^{+195}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 57.8% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+197}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-6)
   (* x (* eps -0.5))
   (if (<= x 480.0)
     1.0
     (if (<= x 1.32e+197) 0.0 (if (<= x 1.35e+297) (* (* x eps) 0.5) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = x * (eps * -0.5);
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 1.32e+197) {
		tmp = 0.0;
	} else if (x <= 1.35e+297) {
		tmp = (x * eps) * 0.5;
	} 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 <= (-9d-6)) then
        tmp = x * (eps * (-0.5d0))
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else if (x <= 1.32d+197) then
        tmp = 0.0d0
    else if (x <= 1.35d+297) then
        tmp = (x * eps) * 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = x * (eps * -0.5);
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 1.32e+197) {
		tmp = 0.0;
	} else if (x <= 1.35e+297) {
		tmp = (x * eps) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-6:
		tmp = x * (eps * -0.5)
	elif x <= 480.0:
		tmp = 1.0
	elif x <= 1.32e+197:
		tmp = 0.0
	elif x <= 1.35e+297:
		tmp = (x * eps) * 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(x * Float64(eps * -0.5));
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 1.32e+197)
		tmp = 0.0;
	elseif (x <= 1.35e+297)
		tmp = Float64(Float64(x * eps) * 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = x * (eps * -0.5);
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 1.32e+197)
		tmp = 0.0;
	elseif (x <= 1.35e+297)
		tmp = (x * eps) * 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-6], N[(x * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 1.32e+197], 0.0, If[LessEqual[x, 1.35e+297], N[(N[(x * eps), $MachinePrecision] * 0.5), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in x around 0 53.6%

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

   5. Taylor expanded in eps around inf 24.3%

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

      1. associate-*r* 24.3%

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

      2. mul-1-neg 24.3%

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

   7. Simplified 24.3%

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

   8. Taylor expanded in eps around 0 24.3%

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

      1. *-commutative 24.3%

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

      2. *-commutative 24.3%

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

      3. associate-*l* 24.3%

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

   10. Simplified 24.3%

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

   if -9.00000000000000023e-6 < x < 480

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around 0 71.6%

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

   if 480 < x < 1.3200000000000001e197 or 1.35e297 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 50.8%

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

   8. Simplified 50.8%

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

   if 1.3200000000000001e197 < x < 1.35e297

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 40.5%

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

   5. Taylor expanded in eps around inf 30.9%

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

      1. associate-*r* 30.9%

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

      2. mul-1-neg 30.9%

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

   7. Simplified 30.9%

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

      1. div-inv 30.9%

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

      2. *-commutative 30.9%

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

      3. add-sqr-sqrt 30.5%

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

      4. sqrt-unprod 71.2%

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

      5. sqr-neg 71.2%

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

      6. sqrt-unprod 26.2%

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

      7. add-sqr-sqrt 26.7%

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

      8. metadata-eval 26.7%

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

   9. Applied egg-rr 26.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+197}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+297}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 60.6% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-6) (* x (* eps -0.5)) (if (<= x 580.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-6) {
		tmp = x * (eps * -0.5);
	} else if (x <= 580.0) {
		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 <= (-9d-6)) then
        tmp = x * (eps * (-0.5d0))
    else if (x <= 580.0d0) 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 <= -9e-6) {
		tmp = x * (eps * -0.5);
	} else if (x <= 580.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-6:
		tmp = x * (eps * -0.5)
	elif x <= 580.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-6)
		tmp = Float64(x * Float64(eps * -0.5));
	elseif (x <= 580.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-6)
		tmp = x * (eps * -0.5);
	elseif (x <= 580.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-6], N[(x * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 580.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000023e-6

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in x around 0 53.6%

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

   5. Taylor expanded in eps around inf 24.3%

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

      1. associate-*r* 24.3%

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

      2. mul-1-neg 24.3%

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

   7. Simplified 24.3%

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

   8. Taylor expanded in eps around 0 24.3%

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

      1. *-commutative 24.3%

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

      2. *-commutative 24.3%

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

      3. associate-*l* 24.3%

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

   10. Simplified 24.3%

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

   if -9.00000000000000023e-6 < x < 580

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in x around 0 71.6%

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

   if 580 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 46.4%

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

   8. Simplified 46.4%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 23.3% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 580.0) 0.5 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 580.0) {
		tmp = 0.5;
	} 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 <= 580.0d0) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 580.0) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 580.0:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 580.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 580.0)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 580.0], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 580

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in eps around 0 59.2%

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

   6. Simplified 59.7%

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

   7. Taylor expanded in x around inf 12.3%

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

   8. Taylor expanded in x around 0 12.2%

      \[\leadsto \frac{\color{blue}{1}}{2} \]

   if 580 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 46.4%

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

   8. Simplified 46.4%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 57.5% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

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

Python

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

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around 0 59.2%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Applied egg-rr 0.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
   7. Step-by-step derivation

      1. div-sub 0.0%

         \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

      2. +-inverses 46.4%

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

   8. Simplified 46.4%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 16.1% 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 75.0%

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

3. Simplified 75.0%

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

4. Taylor expanded in eps around inf 99.2%

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

6. Applied egg-rr 1.6%

   \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3} - {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]
7. Step-by-step derivation

   1. div-sub 1.6%

      \[\leadsto \frac{\color{blue}{\frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)} - \frac{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{3}}{{\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + \left({\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2} + {\left({\left(e^{x}\right)}^{\left(1 + \varepsilon\right)}\right)}^{2}\right)}}}{2} \]

   2. +-inverses 14.2%

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

8. Simplified 14.2%

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

9. Final simplification 14.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023337 
(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))