NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.8% → 99.6%

Time: 18.9s

Alternatives: 9

Speedup: 1.9×

• 37.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{2 \cdot \left(t\_1 + x \cdot t\_1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
          (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps)))))
        (t_1 (exp (- x))))
   (if (<= t_0 0.0) (/ (* 2.0 (+ t_1 (* x t_1))) 2.0) (/ t_0 2.0))))

C

double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	double t_1 = exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (2.0 * (t_1 + (x * t_1))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))
    t_1 = exp(-x)
    if (t_0 <= 0.0d0) then
        tmp = (2.0d0 * (t_1 + (x * t_1))) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	double t_1 = Math.exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (2.0 * (t_1 + (x * t_1))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))
	t_1 = math.exp(-x)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (2.0 * (t_1 + (x * t_1))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps))))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(2.0 * Float64(t_1 + Float64(x * t_1))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	t_1 = exp(-x);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (2.0 * (t_1 + (x * t_1))) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(2.0 * N[(t$95$1 + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 0.0

   1. Initial program 32.5%

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

   3. Simplified 32.5%

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

   5. Taylor expanded in eps around 0 100.0%

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

      1. associate--r+ 100.0%

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

      2. associate-*r* 100.0%

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

      3. mul-1-neg 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. distribute-rgt-out-- 99.9%

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

      7. mul-1-neg 99.9%

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

      8. mul-1-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. distribute-lft-out 100.0%

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

   10. Simplified 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-244}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{1 + x}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-244)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 6e+190)
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (/ (* 2.0 (/ (+ 1.0 x) (exp x))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-244) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 6e+190) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else {
		tmp = (2.0 * ((1.0 + x) / exp(x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-244)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 6d+190) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else
        tmp = (2.0d0 * ((1.0d0 + x) / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-244) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 6e+190) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = (2.0 * ((1.0 + x) / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-244:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 6e+190:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = (2.0 * ((1.0 + x) / math.exp(x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-244)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 6e+190)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-244)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 6e+190)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	else
		tmp = (2.0 * ((1.0 + x) / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-244], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+190], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999999e-244

   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 43.9%

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

   6. Taylor expanded in eps around inf 74.5%

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

      1. neg-mul-1 74.5%

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

      2. distribute-rgt-neg-in 74.5%

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

   8. Simplified 74.5%

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

      1. add-sqr-sqrt 5.0%

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

      2. sqrt-unprod 42.8%

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

      3. sqr-neg 42.8%

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

      4. sqrt-unprod 37.8%

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

      5. add-sqr-sqrt 67.3%

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

      6. sub-neg 67.3%

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

      7. distribute-lft-in 67.3%

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

      8. *-commutative 67.3%

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

      9. *-un-lft-identity 67.3%

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

   10. Applied egg-rr 67.3%

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

       1. *-rgt-identity 67.3%

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

       2. distribute-lft-in 67.3%

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

       3. sub-neg 67.3%

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

   12. Simplified 67.3%

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

   13. Taylor expanded in eps around inf 67.9%

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

       1. associate-*r* 67.9%

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

       2. mul-1-neg 67.9%

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

   15. Simplified 67.9%

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

   if -1.9999999999999999e-244 < x < 5.99999999999999964e190

   1. Initial program 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in eps around inf 75.3%

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

      1. neg-mul-1 75.3%

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

      2. distribute-rgt-neg-in 75.3%

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

   8. Simplified 75.3%

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

   9. Taylor expanded in eps around inf 75.7%

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

       1. *-commutative 75.7%

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

   11. Simplified 75.7%

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

   if 5.99999999999999964e190 < 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%

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

   5. Taylor expanded in eps around 0 63.1%

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

      1. associate--r+ 63.1%

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

      2. associate-*r* 63.1%

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

      3. mul-1-neg 63.1%

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

      4. cancel-sign-sub 63.1%

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

      5. distribute-rgt1-in 63.1%

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

      6. distribute-rgt-out-- 63.1%

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

      7. mul-1-neg 63.1%

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

      8. mul-1-neg 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in x around inf 63.1%

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

      1. distribute-lft-out 63.1%

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

      2. neg-mul-1 63.1%

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

      3. neg-mul-1 63.1%

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

      4. distribute-rgt1-in 63.1%

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

      5. neg-mul-1 63.1%

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

      6. exp-neg 63.1%

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

      7. associate-/l* 63.1%

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

      8. *-rgt-identity 63.1%

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

      9. +-commutative 63.1%

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

   10. Simplified 63.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-244}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{1 + x}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0215:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+189}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.0215)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1.02e+189) (/ (+ 1.0 (exp (* eps x))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.0215) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.02e+189) {
		tmp = (1.0 + exp((eps * 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 <= (-0.0215d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.02d+189) then
        tmp = (1.0d0 + exp((eps * 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 <= -0.0215) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.02e+189) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.0215:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.02e+189:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.0215)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.02e+189)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.0215)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.02e+189)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.0215], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.02e+189], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.021499999999999998

   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 53.0%

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

   6. Taylor expanded in eps around inf 53.0%

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

      1. neg-mul-1 53.0%

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

      2. distribute-rgt-neg-in 53.0%

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

   8. Simplified 53.0%

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

   9. Taylor expanded in eps around 0 94.2%

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

       1. neg-mul-1 94.2%

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

   11. Simplified 94.2%

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

   if -0.021499999999999998 < x < 1.02e189

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around 0 41.7%

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

   6. Taylor expanded in eps around inf 78.8%

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

      1. neg-mul-1 78.8%

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

      2. distribute-rgt-neg-in 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in eps around inf 79.3%

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

       1. *-commutative 79.3%

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

   11. Simplified 79.3%

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

   if 1.02e189 < 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 63.1%

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

      1. mul-1-neg 63.1%

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

      2. mul-1-neg 63.1%

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

      3. rec-exp 63.1%

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

      4. sub-neg 63.1%

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

      5. div-sub 63.1%

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

      6. rec-exp 63.1%

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

      7. mul-1-neg 63.1%

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

      8. +-inverses 63.1%

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

   7. Simplified 63.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0215:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+189}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.3e-245)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 3.1e+190) (/ (+ 1.0 (exp (* eps x))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.3e-245) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 3.1e+190) {
		tmp = (1.0 + exp((eps * 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 <= (-1.3d-245)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 3.1d+190) then
        tmp = (1.0d0 + exp((eps * 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 <= -1.3e-245) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 3.1e+190) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.3e-245:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 3.1e+190:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.3e-245)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 3.1e+190)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.3e-245)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 3.1e+190)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.3e-245], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.1e+190], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000003e-245

   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 43.9%

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

   6. Taylor expanded in eps around inf 74.5%

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

      1. neg-mul-1 74.5%

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

      2. distribute-rgt-neg-in 74.5%

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

   8. Simplified 74.5%

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

      1. add-sqr-sqrt 5.0%

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

      2. sqrt-unprod 42.8%

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

      3. sqr-neg 42.8%

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

      4. sqrt-unprod 37.8%

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

      5. add-sqr-sqrt 67.3%

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

      6. sub-neg 67.3%

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

      7. distribute-lft-in 67.3%

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

      8. *-commutative 67.3%

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

      9. *-un-lft-identity 67.3%

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

   10. Applied egg-rr 67.3%

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

       1. *-rgt-identity 67.3%

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

       2. distribute-lft-in 67.3%

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

       3. sub-neg 67.3%

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

   12. Simplified 67.3%

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

   13. Taylor expanded in eps around inf 67.9%

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

       1. associate-*r* 67.9%

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

       2. mul-1-neg 67.9%

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

   15. Simplified 67.9%

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

   if -1.30000000000000003e-245 < x < 3.1000000000000001e190

   1. Initial program 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in eps around inf 75.3%

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

      1. neg-mul-1 75.3%

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

      2. distribute-rgt-neg-in 75.3%

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

   8. Simplified 75.3%

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

   9. Taylor expanded in eps around inf 75.7%

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

       1. *-commutative 75.7%

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

   11. Simplified 75.7%

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

   if 3.1000000000000001e190 < 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 63.1%

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

      1. mul-1-neg 63.1%

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

      2. mul-1-neg 63.1%

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

      3. rec-exp 63.1%

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

      4. sub-neg 63.1%

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

      5. div-sub 63.1%

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

      6. rec-exp 63.1%

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

      7. mul-1-neg 63.1%

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

      8. +-inverses 63.1%

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

   7. Simplified 63.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-245}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+190}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 150:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 150.0)
   (/ (- 2.0 (* eps x)) 2.0)
   (if (<= x 7e+134) (/ (/ (expm1 x) eps) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 150.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 7e+134) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 150.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 7e+134) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 150.0:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 7e+134:
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 150.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 7e+134)
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 150.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+134], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 150

   1. Initial program 60.8%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in x around 0 62.9%

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

   6. Taylor expanded in eps around 0 67.2%

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

   7. Taylor expanded in x around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. *-commutative 67.2%

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

      3. unsub-neg 67.2%

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

   9. Simplified 67.2%

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

   if 150 < x < 7.00000000000000006e134

   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 40.1%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 1.8%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   8. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-undefine 1.8%

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

      2. div-sub 1.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 38.9%

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

      5. sqr-neg 38.9%

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

      6. sqrt-unprod 38.9%

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

      7. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. div-sub 38.9%

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

       2. expm1-undefine 38.9%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{\varepsilon}}{2} \]

   12. Simplified 38.9%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]

   if 7.00000000000000006e134 < 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 60.2%

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

      1. mul-1-neg 60.2%

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

      2. mul-1-neg 60.2%

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

      3. rec-exp 60.2%

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

      4. sub-neg 60.2%

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

      5. div-sub 60.2%

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

      6. rec-exp 60.2%

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

      7. mul-1-neg 60.2%

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

      8. +-inverses 60.2%

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

   7. Simplified 60.2%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 150:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 470.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 2e+135) (/ (/ (expm1 x) eps) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 470.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2e+135) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 470.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2e+135) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 470.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2e+135:
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 470.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2e+135)
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 470.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+135], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 470

   1. Initial program 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in x around 0 44.5%

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

   6. Taylor expanded in eps around inf 82.0%

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

      1. neg-mul-1 82.0%

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

      2. distribute-rgt-neg-in 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in eps around 0 78.3%

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

       1. neg-mul-1 78.3%

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

   11. Simplified 78.3%

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

   if 470 < x < 1.99999999999999992e135

   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 40.1%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 1.8%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   8. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-undefine 1.8%

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

      2. div-sub 1.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 38.9%

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

      5. sqr-neg 38.9%

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

      6. sqrt-unprod 38.9%

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

      7. add-sqr-sqrt 38.9%

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

   10. Applied egg-rr 38.9%

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

       1. div-sub 38.9%

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

       2. expm1-undefine 38.9%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{\varepsilon}}{2} \]

   12. Simplified 38.9%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]

   if 1.99999999999999992e135 < 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 60.2%

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

      1. mul-1-neg 60.2%

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

      2. mul-1-neg 60.2%

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

      3. rec-exp 60.2%

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

      4. sub-neg 60.2%

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

      5. div-sub 60.2%

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

      6. rec-exp 60.2%

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

      7. mul-1-neg 60.2%

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

      8. +-inverses 60.2%

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

   7. Simplified 60.2%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+135}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 260:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 260.0) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 260.0) {
		tmp = (2.0 - (eps * 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 <= 260.0d0) then
        tmp = (2.0d0 - (eps * 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 <= 260.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 260.0:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 260.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 260.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 260.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 260

   1. Initial program 60.8%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in x around 0 62.9%

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

   6. Taylor expanded in eps around 0 67.2%

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

   7. Taylor expanded in x around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. *-commutative 67.2%

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

      3. unsub-neg 67.2%

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

   9. Simplified 67.2%

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

   if 260 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 49.3%

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

      1. mul-1-neg 49.3%

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

      2. mul-1-neg 49.3%

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

      3. rec-exp 49.3%

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

      4. sub-neg 49.3%

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

      5. div-sub 49.3%

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

      6. rec-exp 49.3%

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

      7. mul-1-neg 49.3%

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

      8. +-inverses 49.3%

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

   7. Simplified 49.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 260:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 3.7e+35) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e+35) {
		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 <= 3.7d+35) 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 <= 3.7e+35) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.7e+35:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.7e+35)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.7e+35)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.7e+35], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.7e35

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in x around 0 61.4%

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

   if 3.7e35 < 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 53.1%

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

      1. mul-1-neg 53.1%

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

      2. mul-1-neg 53.1%

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

      3. rec-exp 53.1%

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

      4. sub-neg 53.1%

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

      5. div-sub 53.1%

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

      6. rec-exp 53.1%

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

      7. mul-1-neg 53.1%

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

      8. +-inverses 53.1%

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

   7. Simplified 53.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 15.8% accurate, 227.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 71.2%

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

3. Simplified 59.3%

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

5. Taylor expanded in eps around 0 14.8%

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

   1. mul-1-neg 14.8%

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

   2. mul-1-neg 14.8%

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

   3. rec-exp 14.8%

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

   4. sub-neg 14.8%

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

   5. div-sub 14.8%

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

   6. rec-exp 14.8%

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

   7. mul-1-neg 14.8%

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

   8. +-inverses 15.0%

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

7. Simplified 15.0%

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

8. Final simplification 15.0%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(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))