NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.1% → 99.6%

Time: 11.3s

Alternatives: 9

Speedup: 2.0×

• 37.9% 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: 74.1% 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

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

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))))))
   (if (<= t_0 0.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 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 tmp;
	if (t_0 <= 0.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 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) :: tmp
    t_0 = ((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 - (1.0d0 / eps)))
    if (t_0 <= 0.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 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 tmp;
	if (t_0 <= 0.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 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)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 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))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 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)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 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]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $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 39.9%

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

      1. div-sub 39.9%

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

      2. +-rgt-identity 39.9%

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

      3. div-sub 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft1-in 100.0%

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

      8. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   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%

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

4. Final simplification 100.0%

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

Alternative 2: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(1 + x\right) \cdot t_0\\ \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + t_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (+ 1.0 x) t_0)))
   (if (<= x -720.0)
     (/ (+ 1.0 t_0) 2.0)
     (if (<= x 1.3e-10)
       (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
       (/ (+ t_1 t_1) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double t_1 = (1.0 + x) * t_0;
	double tmp;
	if (x <= -720.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 1.3e-10) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else {
		tmp = (t_1 + t_1) / 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 = exp(-x)
    t_1 = (1.0d0 + x) * t_0
    if (x <= (-720.0d0)) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if (x <= 1.3d-10) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else
        tmp = (t_1 + t_1) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double t_1 = (1.0 + x) * t_0;
	double tmp;
	if (x <= -720.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 1.3e-10) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = (t_1 + t_1) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-x)
	t_1 = (1.0 + x) * t_0
	tmp = 0
	if x <= -720.0:
		tmp = (1.0 + t_0) / 2.0
	elif x <= 1.3e-10:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	else:
		tmp = (t_1 + t_1) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	t_1 = Float64(Float64(1.0 + x) * t_0)
	tmp = 0.0
	if (x <= -720.0)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (x <= 1.3e-10)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(t_1 + t_1) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	t_1 = (1.0 + x) * t_0;
	tmp = 0.0;
	if (x <= -720.0)
		tmp = (1.0 + t_0) / 2.0;
	elseif (x <= 1.3e-10)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	else
		tmp = (t_1 + t_1) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, -720.0], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.3e-10], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -720

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 42.3%

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

   5. Taylor expanded in eps around inf 42.3%

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

      1. cancel-sign-sub-inv 42.3%

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

      2. metadata-eval 42.3%

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

      3. *-lft-identity 42.3%

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

      4. exp-prod 42.3%

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

      5. +-commutative 42.3%

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

      6. *-lft-identity 42.3%

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

      7. metadata-eval 42.3%

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

      8. cancel-sign-sub-inv 42.3%

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

      9. exp-prod 42.3%

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

      10. *-commutative 42.3%

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

      11. *-commutative 42.3%

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

      12. sub-neg 42.3%

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

      13. mul-1-neg 42.3%

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

      14. remove-double-neg 42.3%

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

      15. associate-*l* 42.3%

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

      16. *-commutative 42.3%

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

      17. distribute-lft-in 42.3%

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

      18. metadata-eval 42.3%

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

      19. mul-1-neg 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -720 < x < 1.29999999999999991e-10

   1. Initial program 57.3%

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

      1. div-sub 57.3%

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

      2. +-rgt-identity 57.3%

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

      3. div-sub 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around 0 47.7%

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

   5. Taylor expanded in eps around inf 90.3%

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

      1. cancel-sign-sub-inv 90.3%

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

      2. metadata-eval 90.3%

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

      3. *-lft-identity 90.3%

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

      4. exp-prod 90.3%

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

      5. +-commutative 90.3%

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

      6. *-lft-identity 90.3%

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

      7. metadata-eval 90.3%

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

      8. cancel-sign-sub-inv 90.3%

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

      9. exp-prod 90.3%

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

      10. *-commutative 90.3%

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

      11. *-commutative 90.3%

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

      12. sub-neg 90.3%

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

      13. mul-1-neg 90.3%

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

      14. remove-double-neg 90.3%

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

      15. associate-*l* 90.3%

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

      16. *-commutative 90.3%

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

      17. distribute-lft-in 90.3%

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

      18. metadata-eval 90.3%

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

      19. mul-1-neg 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in eps around inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. distribute-lft-neg-out 90.3%

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

      3. *-commutative 90.3%

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

   10. Simplified 90.3%

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

   if 1.29999999999999991e-10 < x

   1. Initial program 96.4%

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

      1. div-sub 96.4%

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

      2. +-rgt-identity 96.4%

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

      3. div-sub 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in eps around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-lft1-in 52.1%

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

      3. mul-1-neg 52.1%

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

      4. distribute-lft-out 52.1%

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

      5. mul-1-neg 52.1%

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

      6. *-commutative 52.1%

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

      7. distribute-lft1-in 52.1%

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

      8. mul-1-neg 52.1%

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

   6. Simplified 52.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 3: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -720.0)
     (/ (+ 1.0 t_0) 2.0)
     (if (<= x 1.3e-10)
       (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
       (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) t_0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -720.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 1.3e-10) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * 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) :: tmp
    t_0 = exp(-x)
    if (x <= (-720.0d0)) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if (x <= 1.3d-10) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -720.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 1.3e-10) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -720.0:
		tmp = (1.0 + t_0) / 2.0
	elif x <= 1.3e-10:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	else:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * t_0)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -720.0)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (x <= 1.3e-10)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -720.0)
		tmp = (1.0 + t_0) / 2.0;
	elseif (x <= 1.3e-10)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	else
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -720.0], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.3e-10], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -720

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 42.3%

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

   5. Taylor expanded in eps around inf 42.3%

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

      1. cancel-sign-sub-inv 42.3%

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

      2. metadata-eval 42.3%

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

      3. *-lft-identity 42.3%

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

      4. exp-prod 42.3%

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

      5. +-commutative 42.3%

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

      6. *-lft-identity 42.3%

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

      7. metadata-eval 42.3%

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

      8. cancel-sign-sub-inv 42.3%

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

      9. exp-prod 42.3%

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

      10. *-commutative 42.3%

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

      11. *-commutative 42.3%

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

      12. sub-neg 42.3%

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

      13. mul-1-neg 42.3%

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

      14. remove-double-neg 42.3%

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

      15. associate-*l* 42.3%

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

      16. *-commutative 42.3%

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

      17. distribute-lft-in 42.3%

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

      18. metadata-eval 42.3%

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

      19. mul-1-neg 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -720 < x < 1.29999999999999991e-10

   1. Initial program 57.3%

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

      1. div-sub 57.3%

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

      2. +-rgt-identity 57.3%

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

      3. div-sub 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in x around 0 47.7%

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

   5. Taylor expanded in eps around inf 90.3%

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

      1. cancel-sign-sub-inv 90.3%

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

      2. metadata-eval 90.3%

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

      3. *-lft-identity 90.3%

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

      4. exp-prod 90.3%

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

      5. +-commutative 90.3%

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

      6. *-lft-identity 90.3%

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

      7. metadata-eval 90.3%

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

      8. cancel-sign-sub-inv 90.3%

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

      9. exp-prod 90.3%

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

      10. *-commutative 90.3%

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

      11. *-commutative 90.3%

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

      12. sub-neg 90.3%

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

      13. mul-1-neg 90.3%

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

      14. remove-double-neg 90.3%

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

      15. associate-*l* 90.3%

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

      16. *-commutative 90.3%

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

      17. distribute-lft-in 90.3%

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

      18. metadata-eval 90.3%

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

      19. mul-1-neg 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in eps around inf 90.3%

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

      1. mul-1-neg 90.3%

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

      2. distribute-lft-neg-out 90.3%

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

      3. *-commutative 90.3%

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

   10. Simplified 90.3%

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

   if 1.29999999999999991e-10 < x

   1. Initial program 96.4%

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

      1. div-sub 96.4%

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

      2. +-rgt-identity 96.4%

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

      3. div-sub 96.4%

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

   3. Simplified 96.4%

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

   4. Taylor expanded in eps around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-lft1-in 52.1%

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

      3. mul-1-neg 52.1%

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

      4. distribute-lft-out 52.1%

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

      5. mul-1-neg 52.1%

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

      6. *-commutative 52.1%

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

      7. distribute-lft1-in 52.1%

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

      8. mul-1-neg 52.1%

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

   6. Simplified 52.1%

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

      1. exp-neg 52.0%

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

      2. un-div-inv 52.0%

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

   8. Applied egg-rr 52.0%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 4: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1880:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -720.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1880.0) (/ (+ 1.0 (exp (* eps (- x)))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -720.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1880.0) {
		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 <= (-720.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1880.0d0) 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 <= -720.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1880.0) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -720.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1880.0:
		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 <= -720.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1880.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -720.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1880.0)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -720.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1880.0], 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 < -720

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 42.3%

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

   5. Taylor expanded in eps around inf 42.3%

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

      1. cancel-sign-sub-inv 42.3%

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

      2. metadata-eval 42.3%

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

      3. *-lft-identity 42.3%

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

      4. exp-prod 42.3%

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

      5. +-commutative 42.3%

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

      6. *-lft-identity 42.3%

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

      7. metadata-eval 42.3%

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

      8. cancel-sign-sub-inv 42.3%

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

      9. exp-prod 42.3%

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

      10. *-commutative 42.3%

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

      11. *-commutative 42.3%

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

      12. sub-neg 42.3%

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

      13. mul-1-neg 42.3%

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

      14. remove-double-neg 42.3%

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

      15. associate-*l* 42.3%

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

      16. *-commutative 42.3%

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

      17. distribute-lft-in 42.3%

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

      18. metadata-eval 42.3%

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

      19. mul-1-neg 42.3%

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

   7. Simplified 42.3%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -720 < x < 1880

   1. Initial program 56.8%

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

      1. div-sub 56.8%

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

      2. +-rgt-identity 56.8%

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

      3. div-sub 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in x around 0 46.0%

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

   5. Taylor expanded in eps around inf 88.0%

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

      1. cancel-sign-sub-inv 88.0%

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

      2. metadata-eval 88.0%

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

      3. *-lft-identity 88.0%

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

      4. exp-prod 88.0%

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

      5. +-commutative 88.0%

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

      6. *-lft-identity 88.0%

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

      7. metadata-eval 88.0%

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

      8. cancel-sign-sub-inv 88.0%

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

      9. exp-prod 88.0%

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

      10. *-commutative 88.0%

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

      11. *-commutative 88.0%

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

      12. sub-neg 88.0%

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

      13. mul-1-neg 88.0%

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

      14. remove-double-neg 88.0%

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

      15. associate-*l* 88.0%

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

      16. *-commutative 88.0%

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

      17. distribute-lft-in 88.0%

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

      18. metadata-eval 88.0%

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

      19. mul-1-neg 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in eps around inf 88.6%

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

      1. mul-1-neg 88.6%

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

      2. distribute-lft-neg-out 88.6%

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

      3. *-commutative 88.6%

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

   10. Simplified 88.6%

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

   if 1880 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. div-sub 51.4%

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

      2. rec-exp 51.4%

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

      3. mul-1-neg 51.4%

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

      4. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1880:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 70.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 500.0) (/ (+ 1.0 (exp (- x))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 500.0) {
		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 <= 500.0d0) 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 <= 500.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 500.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 500.0)
		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 <= 500.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 500.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 66.8%

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

      1. div-sub 66.8%

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

      2. +-rgt-identity 66.8%

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

      3. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in x around 0 45.2%

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

   5. Taylor expanded in eps around inf 77.4%

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

      1. cancel-sign-sub-inv 77.4%

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

      2. metadata-eval 77.4%

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

      3. *-lft-identity 77.4%

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

      4. exp-prod 77.4%

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

      5. +-commutative 77.4%

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

      6. *-lft-identity 77.4%

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

      7. metadata-eval 77.4%

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

      8. cancel-sign-sub-inv 77.4%

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

      9. exp-prod 77.4%

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

      10. *-commutative 77.4%

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

      11. *-commutative 77.4%

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

      12. sub-neg 77.4%

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

      13. mul-1-neg 77.4%

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

      14. remove-double-neg 77.4%

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

      15. associate-*l* 77.4%

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

      16. *-commutative 77.4%

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

      17. distribute-lft-in 77.4%

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

      18. metadata-eval 77.4%

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

      19. mul-1-neg 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in eps around 0 78.7%

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

      1. neg-mul-1 78.7%

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

   10. Simplified 78.7%

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

   if 500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. div-sub 51.4%

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

      2. rec-exp 51.4%

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

      3. mul-1-neg 51.4%

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

      4. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 60.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 25:\\ \;\;\;\;\frac{x \cdot \left(\varepsilon + -1\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 25.0) (/ (+ (* x (+ eps -1.0)) 2.0) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 25.0) {
		tmp = ((x * (eps + -1.0)) + 2.0) / 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 <= 25.0d0) then
        tmp = ((x * (eps + (-1.0d0))) + 2.0d0) / 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 <= 25.0) {
		tmp = ((x * (eps + -1.0)) + 2.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 25.0:
		tmp = ((x * (eps + -1.0)) + 2.0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 25.0)
		tmp = Float64(Float64(Float64(x * Float64(eps + -1.0)) + 2.0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 25.0)
		tmp = ((x * (eps + -1.0)) + 2.0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 25.0], N[(N[(N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 25

   1. Initial program 66.8%

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

      1. div-sub 66.8%

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

      2. +-rgt-identity 66.8%

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

      3. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in x around 0 45.2%

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

   5. Taylor expanded in eps around inf 77.4%

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

      1. *-commutative 77.4%

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

      2. sub-neg 77.4%

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

      3. mul-1-neg 77.4%

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

      4. *-commutative 77.4%

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

      5. +-commutative 77.4%

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

      6. mul-1-neg 77.4%

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

      7. *-commutative 77.4%

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

      8. mul-1-neg 77.4%

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

      9. sub-neg 77.4%

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

   7. Simplified 77.4%

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

   8. Taylor expanded in x around 0 63.1%

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

   if 25 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. div-sub 51.4%

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

      2. rec-exp 51.4%

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

      3. mul-1-neg 51.4%

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

      4. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 25:\\ \;\;\;\;\frac{x \cdot \left(\varepsilon + -1\right) + 2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 60.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.0055) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.0055:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.0055)
		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 <= 0.0055)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.0055], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054999999999999997

   1. Initial program 66.6%

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

      1. div-sub 66.6%

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

      2. +-rgt-identity 66.6%

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

      3. div-sub 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in x around 0 45.4%

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

   5. Taylor expanded in eps around inf 77.8%

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

      1. cancel-sign-sub-inv 77.8%

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

      2. metadata-eval 77.8%

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

      3. *-lft-identity 77.8%

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

      4. exp-prod 77.8%

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

      5. +-commutative 77.8%

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

      6. *-lft-identity 77.8%

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

      7. metadata-eval 77.8%

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

      8. cancel-sign-sub-inv 77.8%

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

      9. exp-prod 77.8%

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

      10. *-commutative 77.8%

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

      11. *-commutative 77.8%

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

      12. sub-neg 77.8%

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

      13. mul-1-neg 77.8%

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

      14. remove-double-neg 77.8%

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

      15. associate-*l* 77.8%

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

      16. *-commutative 77.8%

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

      17. distribute-lft-in 77.8%

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

      18. metadata-eval 77.8%

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

      19. mul-1-neg 77.8%

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

   7. Simplified 77.8%

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

   8. Taylor expanded in eps around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. distribute-lft-neg-out 78.3%

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

      3. *-commutative 78.3%

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

   10. Simplified 78.3%

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

   11. Taylor expanded in x around 0 61.3%

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

       1. +-commutative 61.3%

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

       2. associate-*r* 61.3%

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

       3. mul-1-neg 61.3%

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

       4. cancel-sign-sub-inv 61.3%

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

   13. Simplified 61.3%

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

   if 0.0054999999999999997 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 50.8%

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

      1. div-sub 50.8%

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

      2. rec-exp 50.8%

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

      3. mul-1-neg 50.8%

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

      4. +-inverses 50.8%

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

   6. Simplified 50.8%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 57.6% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 495:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 495.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 495.0) {
		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 <= 495.0d0) 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 <= 495.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 495.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 495.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 495.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 495.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 495

   1. Initial program 66.8%

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

      1. div-sub 66.8%

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

      2. +-rgt-identity 66.8%

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

      3. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in x around 0 56.7%

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

   if 495 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. div-sub 51.4%

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

      2. rec-exp 51.4%

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

      3. mul-1-neg 51.4%

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

      4. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 495:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 16.4% 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 76.5%

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

3. Simplified 72.2%

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

4. Taylor expanded in eps around 0 16.5%

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

   1. div-sub 16.5%

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

   2. rec-exp 16.5%

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

   3. mul-1-neg 16.5%

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

   4. +-inverses 16.8%

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

6. Simplified 16.8%

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

7. Final simplification 16.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023182 
(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))