NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.9% → 98.8%

Time: 11.8s

Alternatives: 14

Speedup: 1.1×

• 37.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.9760765541959742e+103
  2. eps = -2.1666412448371755e-275

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 9.2 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2 - x \cdot \varepsilon}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 9.2e-264)
   0.0
   (if (<= eps 2e-5)
     (/ (* eps (+ x (/ (- 2.0 (* x eps)) eps))) 2.0)
     (/ (+ (exp (* x (- eps))) (exp (* x eps))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 9.2e-264) {
		tmp = 0.0;
	} else if (eps <= 2e-5) {
		tmp = (eps * (x + ((2.0 - (x * eps)) / eps))) / 2.0;
	} else {
		tmp = (exp((x * -eps)) + 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 (eps <= 9.2d-264) then
        tmp = 0.0d0
    else if (eps <= 2d-5) then
        tmp = (eps * (x + ((2.0d0 - (x * eps)) / eps))) / 2.0d0
    else
        tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 9.2e-264) {
		tmp = 0.0;
	} else if (eps <= 2e-5) {
		tmp = (eps * (x + ((2.0 - (x * eps)) / eps))) / 2.0;
	} else {
		tmp = (Math.exp((x * -eps)) + Math.exp((x * eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 9.2e-264:
		tmp = 0.0
	elif eps <= 2e-5:
		tmp = (eps * (x + ((2.0 - (x * eps)) / eps))) / 2.0
	else:
		tmp = (math.exp((x * -eps)) + math.exp((x * eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 9.2e-264)
		tmp = 0.0;
	elseif (eps <= 2e-5)
		tmp = Float64(Float64(eps * Float64(x + Float64(Float64(2.0 - Float64(x * eps)) / eps))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + exp(Float64(x * eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 9.2e-264)
		tmp = 0.0;
	elseif (eps <= 2e-5)
		tmp = (eps * (x + ((2.0 - (x * eps)) / eps))) / 2.0;
	else
		tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 9.2e-264], 0.0, If[LessEqual[eps, 2e-5], N[(N[(eps * N[(x + N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 9.20000000000000046e-264

   1. Initial program 73.4%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in eps around 0 18.6%

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

   6. Taylor expanded in x around 0 18.8%

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

   if 9.20000000000000046e-264 < eps < 2.00000000000000016e-5

   1. Initial program 32.6%

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

   3. Simplified 32.6%

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

   5. Taylor expanded in x around 0 7.0%

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

   6. Taylor expanded in eps around inf 74.4%

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

   8. Simplified 74.4%

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

   9. Taylor expanded in x around 0 74.2%

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

       1. distribute-rgt-in 74.2%

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

       2. *-lft-identity 74.2%

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

       3. distribute-lft-out 74.2%

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

       4. associate-*r* 74.2%

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

       5. distribute-rgt-in 74.2%

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

       6. mul-1-neg 74.2%

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

       7. sub-neg 74.2%

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

   11. Simplified 74.2%

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

   12. Taylor expanded in eps around inf 89.2%

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

       1. associate-*r* 89.2%

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

       2. neg-mul-1 89.2%

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

   14. Simplified 89.2%

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

   if 2.00000000000000016e-5 < eps

   1. Initial program 100.0%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 9.2 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2 - x \cdot \varepsilon}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Taylor expanded in eps around inf 88.9%

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

   1. mul-1-neg 88.9%

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

   2. *-commutative 88.9%

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

   3. distribute-rgt-neg-in 88.9%

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

8. Simplified 88.9%

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

9. Final simplification 88.9%

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

Alternative 4: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-277}:\\ \;\;\;\;\frac{x \cdot \left(\varepsilon + \frac{\varepsilon \cdot \left(\frac{1}{\varepsilon} + \frac{e^{x \cdot \left(-\varepsilon\right)}}{\varepsilon}\right)}{x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.7e-277)
   (/
    (* x (+ eps (/ (* eps (+ (/ 1.0 eps) (/ (exp (* x (- eps))) eps))) x)))
    2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e-277) {
		tmp = (x * (eps + ((eps * ((1.0 / eps) + (exp((x * -eps)) / eps))) / x))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-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 <= 3.7d-277) then
        tmp = (x * (eps + ((eps * ((1.0d0 / eps) + (exp((x * -eps)) / eps))) / x))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e-277) {
		tmp = (x * (eps + ((eps * ((1.0 / eps) + (Math.exp((x * -eps)) / eps))) / x))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.7e-277:
		tmp = (x * (eps + ((eps * ((1.0 / eps) + (math.exp((x * -eps)) / eps))) / x))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.7e-277)
		tmp = Float64(Float64(x * Float64(eps + Float64(Float64(eps * Float64(Float64(1.0 / eps) + Float64(exp(Float64(x * Float64(-eps))) / eps))) / x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.7e-277)
		tmp = (x * (eps + ((eps * ((1.0 / eps) + (exp((x * -eps)) / eps))) / x))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.7e-277], N[(N[(x * N[(eps + N[(N[(eps * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.69999999999999985e-277

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in x around 0 39.0%

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

   6. Taylor expanded in eps around inf 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in x around inf 83.4%

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

   10. Taylor expanded in eps around inf 83.9%

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

       1. mul-1-neg 99.2%

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

       2. *-commutative 99.2%

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

       3. distribute-rgt-neg-in 99.2%

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

   12. Simplified 83.9%

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

   if 3.69999999999999985e-277 < x

   1. Initial program 80.5%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. *-commutative 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in x around 0 48.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-277}:\\ \;\;\;\;\frac{x \cdot \left(\varepsilon + \frac{\varepsilon \cdot \left(\frac{1}{\varepsilon} + \frac{e^{x \cdot \left(-\varepsilon\right)}}{\varepsilon}\right)}{x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 4e-277)
   (/ (* eps (+ x (/ (+ 1.0 (exp (* x (- -1.0 eps)))) eps))) 2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4e-277) {
		tmp = (eps * (x + ((1.0 + exp((x * (-1.0 - eps)))) / eps))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-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 <= 4d-277) then
        tmp = (eps * (x + ((1.0d0 + exp((x * ((-1.0d0) - eps)))) / eps))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 4e-277) {
		tmp = (eps * (x + ((1.0 + Math.exp((x * (-1.0 - eps)))) / eps))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 4e-277:
		tmp = (eps * (x + ((1.0 + math.exp((x * (-1.0 - eps)))) / eps))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4e-277)
		tmp = Float64(Float64(eps * Float64(x + Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / eps))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4e-277)
		tmp = (eps * (x + ((1.0 + exp((x * (-1.0 - eps)))) / eps))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4e-277], N[(N[(eps * N[(x + N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999988e-277

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in x around 0 39.0%

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

   6. Taylor expanded in eps around inf 83.6%

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

   8. Simplified 83.6%

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

   9. Taylor expanded in x around 0 83.6%

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

       1. distribute-rgt-in 83.6%

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

       2. *-lft-identity 83.6%

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

       3. distribute-lft-out 83.6%

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

       4. associate-*r* 83.6%

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

       5. distribute-rgt-in 83.6%

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

       6. mul-1-neg 83.6%

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

       7. sub-neg 83.6%

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

   11. Simplified 83.6%

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

   if 3.99999999999999988e-277 < x

   1. Initial program 80.5%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. *-commutative 79.6%

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

      3. distribute-rgt-neg-in 79.6%

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

   8. Simplified 79.6%

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

   9. Taylor expanded in x around 0 48.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-93)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x -6.5e-115)
     (/ (/ (+ x (* eps (- 2.0 (* x eps)))) eps) 2.0)
     (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-93) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -6.5e-115) {
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-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 <= (-5.2d-93)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= (-6.5d-115)) then
        tmp = ((x + (eps * (2.0d0 - (x * eps)))) / eps) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-93) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= -6.5e-115) {
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e-93:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= -6.5e-115:
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-93)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -6.5e-115)
		tmp = Float64(Float64(Float64(x + Float64(eps * Float64(2.0 - Float64(x * eps)))) / eps) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e-93)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= -6.5e-115)
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-93], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -6.5e-115], N[(N[(N[(x + N[(eps * N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999997e-93

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in x around 0 44.8%

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

   6. Taylor expanded in eps around inf 66.5%

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

      1. mul-1-neg 66.5%

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

      2. neg-mul-1 66.5%

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

      3. distribute-rgt-neg-in 66.5%

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

      4. distribute-neg-in 66.5%

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

      5. metadata-eval 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in eps around 0 90.1%

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

       1. neg-mul-1 90.1%

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

   11. Simplified 90.1%

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

   if -5.1999999999999997e-93 < x < -6.50000000000000033e-115

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in x around 0 17.6%

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

   7. Taylor expanded in eps around 0 33.3%

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

      1. mul-1-neg 33.3%

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

      2. *-commutative 33.3%

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

      3. unsub-neg 33.3%

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

      4. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   if -6.50000000000000033e-115 < x

   1. Initial program 71.0%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. distribute-rgt-neg-in 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in x around 0 60.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.8e-93)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x -6.6e-117)
     (/ (/ (+ x (* eps (- 2.0 (* x eps)))) eps) 2.0)
     (/ (+ 1.0 (exp (* x eps))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.8e-93) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -6.6e-117) {
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 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 <= (-5.8d-93)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= (-6.6d-117)) then
        tmp = ((x + (eps * (2.0d0 - (x * eps)))) / eps) / 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 <= -5.8e-93) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= -6.6e-117) {
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.8e-93:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= -6.6e-117:
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0
	else:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.8e-93)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -6.6e-117)
		tmp = Float64(Float64(Float64(x + Float64(eps * Float64(2.0 - Float64(x * eps)))) / eps) / 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 <= -5.8e-93)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= -6.6e-117)
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	else
		tmp = (1.0 + exp((x * eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.8e-93], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -6.6e-117], N[(N[(N[(x + N[(eps * N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999997e-93

   1. Initial program 76.5%

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

   3. Simplified 76.5%

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

   5. Taylor expanded in x around 0 44.8%

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

   6. Taylor expanded in eps around inf 66.5%

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

      1. mul-1-neg 66.5%

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

      2. neg-mul-1 66.5%

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

      3. distribute-rgt-neg-in 66.5%

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

      4. distribute-neg-in 66.5%

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

      5. metadata-eval 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in eps around 0 90.1%

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

       1. neg-mul-1 90.1%

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

   11. Simplified 90.1%

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

   if -5.7999999999999997e-93 < x < -6.6000000000000003e-117

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in x around 0 17.6%

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

   7. Taylor expanded in eps around 0 33.3%

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

      1. mul-1-neg 33.3%

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

      2. *-commutative 33.3%

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

      3. unsub-neg 33.3%

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

      4. *-commutative 33.3%

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

   9. Simplified 33.3%

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

   if -6.6000000000000003e-117 < x

   1. Initial program 71.0%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. distribute-rgt-neg-in 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in eps around inf 80.9%

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

       1. *-commutative 80.9%

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

   11. Simplified 80.9%

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

   12. Taylor expanded in x around 0 60.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.2e-239)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.2e-239) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-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 <= (-4.2d-239)) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.2e-239) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.2e-239:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.2e-239)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.2e-239)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.2e-239], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000004e-239

   1. Initial program 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in x around 0 42.5%

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

   6. Taylor expanded in eps around inf 72.9%

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

      1. mul-1-neg 72.9%

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

      2. neg-mul-1 72.9%

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

      3. distribute-rgt-neg-in 72.9%

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

      4. distribute-neg-in 72.9%

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

      5. metadata-eval 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in x around 0 72.9%

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

       1. distribute-rgt-in 79.7%

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

       2. *-lft-identity 79.7%

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

       3. distribute-lft-out 79.7%

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

       4. associate-*r* 79.7%

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

       5. distribute-rgt-in 79.7%

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

       6. mul-1-neg 79.7%

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

       7. sub-neg 79.7%

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

   11. Simplified 72.9%

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

   12. Taylor expanded in eps around inf 73.5%

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

       1. associate-*r* 55.5%

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

       2. neg-mul-1 55.5%

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

   14. Simplified 73.5%

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

   if -4.2000000000000004e-239 < x

   1. Initial program 76.1%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 82.7%

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

      1. mul-1-neg 82.7%

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

      2. *-commutative 82.7%

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

      3. distribute-rgt-neg-in 82.7%

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

   8. Simplified 82.7%

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

   9. Taylor expanded in x around 0 56.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500000000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 500000000000.0) (/ (+ 1.0 (exp (- x))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 500000000000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 500000000000.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 500000000000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 500000000000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 500000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 500000000000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 500000000000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5e11

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in eps around inf 78.4%

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

      1. mul-1-neg 78.4%

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

      2. neg-mul-1 78.4%

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

      3. distribute-rgt-neg-in 78.4%

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

      4. distribute-neg-in 78.4%

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

      5. metadata-eval 78.4%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in eps around 0 79.5%

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

       1. neg-mul-1 79.5%

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

   11. Simplified 79.5%

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

   if 5e11 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.8%

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

   6. Taylor expanded in x around 0 50.8%

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

Alternative 10: 62.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 500000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.004)
   (/ (/ (+ x (* eps (- 2.0 (* x eps)))) eps) 2.0)
   (if (<= x 500000000000.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.004) {
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	} else if (x <= 500000000000.0) {
		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 <= (-0.004d0)) then
        tmp = ((x + (eps * (2.0d0 - (x * eps)))) / eps) / 2.0d0
    else if (x <= 500000000000.0d0) 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 <= -0.004) {
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	} else if (x <= 500000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.004:
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0
	elif x <= 500000000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.004)
		tmp = Float64(Float64(Float64(x + Float64(eps * Float64(2.0 - Float64(x * eps)))) / eps) / 2.0);
	elseif (x <= 500000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.004)
		tmp = ((x + (eps * (2.0 - (x * eps)))) / eps) / 2.0;
	elseif (x <= 500000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.004], N[(N[(N[(x + N[(eps * N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 500000000000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0040000000000000001

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 59.4%

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

   6. Taylor expanded in x around 0 32.5%

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

   7. Taylor expanded in eps around 0 51.6%

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

      1. mul-1-neg 51.6%

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

      2. *-commutative 51.6%

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

      3. unsub-neg 51.6%

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

      4. *-commutative 51.6%

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

   9. Simplified 51.6%

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

   if -0.0040000000000000001 < x < 5e11

   1. Initial program 51.2%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in x around 0 37.1%

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

   6. Taylor expanded in eps around inf 84.9%

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

   8. Simplified 84.9%

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

   9. Taylor expanded in x around 0 72.6%

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

       1. distribute-rgt-in 72.6%

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

       2. *-lft-identity 72.6%

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

       3. distribute-lft-out 72.6%

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

       4. associate-*r* 72.6%

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

       5. distribute-rgt-in 72.6%

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

       6. mul-1-neg 72.6%

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

       7. sub-neg 72.6%

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

   11. Simplified 72.6%

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

   12. Taylor expanded in x around 0 73.6%

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

   if 5e11 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.8%

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

   6. Taylor expanded in x around 0 50.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.004:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - x \cdot \varepsilon\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 500000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.4e-9) (/ (+ 2.0 (* x (- -1.0 eps))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.4e-9) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 3.4d-9) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.4e-9) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.4e-9:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.4e-9)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.4e-9)
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.4e-9], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.3999999999999998e-9

   1. Initial program 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in x around 0 41.8%

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

   6. Taylor expanded in eps around inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. neg-mul-1 79.8%

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

      3. distribute-rgt-neg-in 79.8%

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

      4. distribute-neg-in 79.8%

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

      5. metadata-eval 79.8%

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

   8. Simplified 79.8%

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

   9. Taylor expanded in x around 0 65.0%

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

       1. distribute-rgt-in 65.0%

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

       2. *-lft-identity 65.0%

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

       3. distribute-lft-out 65.0%

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

       4. associate-*r* 65.0%

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

       5. distribute-rgt-in 65.0%

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

       6. mul-1-neg 65.0%

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

       7. sub-neg 65.0%

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

   11. Simplified 65.0%

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

   if 3.3999999999999998e-9 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.0%

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

   6. Taylor expanded in x around 0 47.0%

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

Alternative 12: 58.0% accurate, 22.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 2.0) (/ (- 2.0 x) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 - x) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 62.0%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around inf 99.4%

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

      1. mul-1-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 58.2%

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

       1. *-commutative 58.2%

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

       2. distribute-rgt1-in 58.2%

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

       3. metadata-eval 58.2%

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

       4. mul0-lft 58.2%

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

       5. metadata-eval 58.2%

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

       6. neg-mul-1 58.2%

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

       7. unsub-neg 58.2%

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

   11. Simplified 58.2%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 48.8%

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

   6. Taylor expanded in x around 0 48.8%

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

Alternative 13: 57.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 500000000000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 500000000000.0) {
		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 <= 500000000000.0d0) 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 <= 500000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 500000000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 500000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 500000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 500000000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5e11

   1. Initial program 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in x around 0 38.6%

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

   6. Taylor expanded in eps around inf 82.3%

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

   8. Simplified 82.3%

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

   9. Taylor expanded in x around 0 63.0%

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

       1. distribute-rgt-in 63.0%

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

       2. *-lft-identity 63.0%

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

       3. distribute-lft-out 63.0%

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

       4. associate-*r* 63.0%

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

       5. distribute-rgt-in 63.0%

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

       6. mul-1-neg 63.0%

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

       7. sub-neg 63.0%

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

   11. Simplified 63.0%

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

   12. Taylor expanded in x around 0 57.1%

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

   if 5e11 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.8%

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

   6. Taylor expanded in x around 0 50.8%

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

Alternative 14: 16.0% 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 73.1%

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

3. Simplified 64.1%

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

5. Taylor expanded in eps around 0 15.8%

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

6. Taylor expanded in x around 0 16.0%

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

Reproduce

herbie shell --seed 2024186 
(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))