NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 77.3%

Time: 31.1s

Alternatives: 11

Speedup: 0.5×

• 37.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% 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: 77.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = exp((x * (eps + -1.0)));
	double t_1 = exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps));
	double tmp;
	if ((t_1 - (t_0 * (-1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp((eps * x))) + 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 * (eps + (-1.0d0))))
    t_1 = exp((x * ((-1.0d0) - eps))) * (1.0d0 - (1.0d0 / eps))
    if ((t_1 - (t_0 * ((-1.0d0) + ((-1.0d0) / eps)))) <= 0.0d0) then
        tmp = t_0 / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp((eps * x))) + 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 * (eps + -1.0)));
	double t_1 = Math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps));
	double tmp;
	if ((t_1 - (t_0 * (-1.0 + (-1.0 / eps)))) <= 0.0) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp((eps * x))) + t_1) / 2.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[(t$95$0 * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / 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 36.0%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in eps around inf 12.0%

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

      1. mul-1-neg 12.0%

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

      2. distribute-lft-neg-out 12.0%

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

      3. *-commutative 12.0%

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

   7. Simplified 12.0%

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

   8. Taylor expanded in eps around 0 2.8%

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

   9. Taylor expanded in eps around inf 46.3%

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

       1. mul-1-neg 46.3%

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

       2. distribute-rgt-neg-in 46.3%

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

   11. Simplified 46.3%

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

   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. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 75.9%

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

Alternative 2: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(\varepsilon + -1\right)}\\ t_1 := 1 - \frac{1}{\varepsilon}\\ \mathbf{if}\;\varepsilon \leq 6 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_0}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot t\_0 + e^{-x} \cdot t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot t\_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps -1.0)))) (t_1 (- 1.0 (/ 1.0 eps))))
   (if (<= eps 6e-17)
     (/ t_0 2.0)
     (if (<= eps 2.45e+14)
       (/ (+ (* (+ 1.0 (/ 1.0 eps)) t_0) (* (exp (- x)) t_1)) 2.0)
       (/ (+ (exp (* eps x)) (* (exp (* x (- -1.0 eps))) t_1)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (eps + -1.0)));
	double t_1 = 1.0 - (1.0 / eps);
	double tmp;
	if (eps <= 6e-17) {
		tmp = t_0 / 2.0;
	} else if (eps <= 2.45e+14) {
		tmp = (((1.0 + (1.0 / eps)) * t_0) + (exp(-x) * t_1)) / 2.0;
	} else {
		tmp = (exp((eps * x)) + (exp((x * (-1.0 - eps))) * 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 * (eps + (-1.0d0))))
    t_1 = 1.0d0 - (1.0d0 / eps)
    if (eps <= 6d-17) then
        tmp = t_0 / 2.0d0
    else if (eps <= 2.45d+14) then
        tmp = (((1.0d0 + (1.0d0 / eps)) * t_0) + (exp(-x) * t_1)) / 2.0d0
    else
        tmp = (exp((eps * x)) + (exp((x * ((-1.0d0) - eps))) * 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 * (eps + -1.0)));
	double t_1 = 1.0 - (1.0 / eps);
	double tmp;
	if (eps <= 6e-17) {
		tmp = t_0 / 2.0;
	} else if (eps <= 2.45e+14) {
		tmp = (((1.0 + (1.0 / eps)) * t_0) + (Math.exp(-x) * t_1)) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + (Math.exp((x * (-1.0 - eps))) * t_1)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * (eps + -1.0)))
	t_1 = 1.0 - (1.0 / eps)
	tmp = 0
	if eps <= 6e-17:
		tmp = t_0 / 2.0
	elif eps <= 2.45e+14:
		tmp = (((1.0 + (1.0 / eps)) * t_0) + (math.exp(-x) * t_1)) / 2.0
	else:
		tmp = (math.exp((eps * x)) + (math.exp((x * (-1.0 - eps))) * t_1)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(eps + -1.0)))
	t_1 = Float64(1.0 - Float64(1.0 / eps))
	tmp = 0.0
	if (eps <= 6e-17)
		tmp = Float64(t_0 / 2.0);
	elseif (eps <= 2.45e+14)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * t_0) + Float64(exp(Float64(-x)) * t_1)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * t_1)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * (eps + -1.0)));
	t_1 = 1.0 - (1.0 / eps);
	tmp = 0.0;
	if (eps <= 6e-17)
		tmp = t_0 / 2.0;
	elseif (eps <= 2.45e+14)
		tmp = (((1.0 + (1.0 / eps)) * t_0) + (exp(-x) * t_1)) / 2.0;
	else
		tmp = (exp((eps * x)) + (exp((x * (-1.0 - eps))) * t_1)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 6e-17], N[(t$95$0 / 2.0), $MachinePrecision], If[LessEqual[eps, 2.45e+14], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Exp[(-x)], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 6.00000000000000012e-17

   1. Initial program 58.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in eps around inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. distribute-lft-neg-out 43.5%

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

      3. *-commutative 43.5%

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

   7. Simplified 43.5%

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

   8. Taylor expanded in eps around 0 18.9%

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

   9. Taylor expanded in eps around inf 46.8%

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

       1. mul-1-neg 46.8%

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

       2. distribute-rgt-neg-in 46.8%

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

   11. Simplified 46.8%

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

   if 6.00000000000000012e-17 < eps < 2.45e14

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   if 2.45e14 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6 \cdot 10^{-17}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(\varepsilon + -1\right)}\\ \mathbf{if}\;\varepsilon \leq 0.5:\\ \;\;\;\;\frac{t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot t\_0 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps -1.0)))))
   (if (<= eps 0.5)
     (/ t_0 2.0)
     (/ (+ (* (+ 1.0 (/ 1.0 eps)) t_0) (exp (* x (- eps)))) 2.0))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (eps + -1.0)));
	double tmp;
	if (eps <= 0.5) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * t_0) + exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps + (-1.0d0))))
    if (eps <= 0.5d0) then
        tmp = t_0 / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps)) * t_0) + exp((x * -eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * (eps + -1.0)));
	double tmp;
	if (eps <= 0.5) {
		tmp = t_0 / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * t_0) + Math.exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * (eps + -1.0)))
	tmp = 0
	if eps <= 0.5:
		tmp = t_0 / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps)) * t_0) + math.exp((x * -eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(eps + -1.0)))
	tmp = 0.0
	if (eps <= 0.5)
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * t_0) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * (eps + -1.0)));
	tmp = 0.0;
	if (eps <= 0.5)
		tmp = t_0 / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps)) * t_0) + exp((x * -eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps, 0.5], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.5

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in eps around inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. distribute-lft-neg-out 44.4%

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

      3. *-commutative 44.4%

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

   7. Simplified 44.4%

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

   8. Taylor expanded in eps around 0 18.9%

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

   9. Taylor expanded in eps around inf 46.3%

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

       1. mul-1-neg 46.3%

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

       2. distribute-rgt-neg-in 46.3%

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

   11. Simplified 46.3%

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

   if 0.5 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-neg-out 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. mul-1-neg 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 61.2%

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

Alternative 4: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in eps around inf 44.4%

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

      1. mul-1-neg 44.4%

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

      2. distribute-lft-neg-out 44.4%

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

      3. *-commutative 44.4%

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

   7. Simplified 44.4%

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

   8. Taylor expanded in eps around 0 18.9%

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

   9. Taylor expanded in eps around inf 46.3%

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

       1. mul-1-neg 46.3%

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

       2. distribute-rgt-neg-in 46.3%

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

   11. Simplified 46.3%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 97.8%

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

4. Final simplification 61.2%

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

Alternative 5: 51.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.4e-176)
   (/
    (+ (+ 1.0 (/ 1.0 eps)) (* (exp (* x (- -1.0 eps))) (- 1.0 (/ 1.0 eps))))
    2.0)
   (/ (exp (* x (+ eps -1.0))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.4e-176) {
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0;
	} else {
		tmp = exp((x * (eps + -1.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.4d-176) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 - (1.0d0 / eps)))) / 2.0d0
    else
        tmp = exp((x * (eps + (-1.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.4e-176) {
		tmp = ((1.0 + (1.0 / eps)) + (Math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0;
	} else {
		tmp = Math.exp((x * (eps + -1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.4e-176:
		tmp = ((1.0 + (1.0 / eps)) + (math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0
	else:
		tmp = math.exp((x * (eps + -1.0))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.4e-176)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 - Float64(1.0 / eps)))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps + -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.4e-176)
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) / 2.0;
	else
		tmp = exp((x * (eps + -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.4e-176], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.40000000000000006e-176

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in eps around inf 63.7%

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

      1. *-commutative 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in x around 0 45.9%

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

   if 2.40000000000000006e-176 < x

   1. Initial program 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in eps around inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. distribute-lft-neg-out 55.5%

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

      3. *-commutative 55.5%

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

   7. Simplified 55.5%

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

   8. Taylor expanded in eps around 0 22.6%

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

   9. Taylor expanded in eps around inf 63.0%

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

       1. mul-1-neg 63.0%

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

       2. distribute-rgt-neg-in 63.0%

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

   11. Simplified 63.0%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{-176}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, 1, 1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-100)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 1.55e-177)
     (/ (fma (+ 1.0 (/ 1.0 eps)) 1.0 (- 1.0 (/ 1.0 eps))) 2.0)
     (/ (exp (* x (+ eps -1.0))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-100) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 1.55e-177) {
		tmp = fma((1.0 + (1.0 / eps)), 1.0, (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = exp((x * (eps + -1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-100)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 1.55e-177)
		tmp = Float64(fma(Float64(1.0 + Float64(1.0 / eps)), 1.0, Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps + -1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-100], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.55e-177], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0 + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000001e-100

   1. Initial program 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in eps around inf 77.6%

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

      1. mul-1-neg 77.6%

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

      2. distribute-lft-neg-out 77.6%

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

      3. *-commutative 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in eps around 0 36.5%

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

   9. Taylor expanded in eps around 0 25.7%

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

       1. expm1-define 25.8%

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

       2. mul-1-neg 25.8%

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

   11. Simplified 25.8%

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

   if -4.0000000000000001e-100 < x < 1.55000000000000009e-177

   1. Initial program 53.7%

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

   3. Simplified 47.7%

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

   5. Taylor expanded in x around 0 47.7%

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

   6. Taylor expanded in x around 0 41.7%

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

   if 1.55000000000000009e-177 < x

   1. Initial program 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in eps around inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. distribute-lft-neg-out 55.5%

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

      3. *-commutative 55.5%

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

   7. Simplified 55.5%

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

   8. Taylor expanded in eps around 0 22.6%

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

   9. Taylor expanded in eps around inf 63.0%

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

       1. mul-1-neg 63.0%

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

       2. distribute-rgt-neg-in 63.0%

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

   11. Simplified 63.0%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, 1, 1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, 1, 1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.6e-224)
   (/ (- (/ -1.0 eps) (* (exp (* (- 1.0 eps) x)) (+ -1.0 (/ -1.0 eps)))) 2.0)
   (if (<= x 8.8e-175)
     (/ (fma (+ 1.0 (/ 1.0 eps)) 1.0 (- 1.0 (/ 1.0 eps))) 2.0)
     (/ (exp (* x (+ eps -1.0))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.6e-224) {
		tmp = ((-1.0 / eps) - (exp(((1.0 - eps) * x)) * (-1.0 + (-1.0 / eps)))) / 2.0;
	} else if (x <= 8.8e-175) {
		tmp = fma((1.0 + (1.0 / eps)), 1.0, (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = exp((x * (eps + -1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.6e-224)
		tmp = Float64(Float64(Float64(-1.0 / eps) - Float64(exp(Float64(Float64(1.0 - eps) * x)) * Float64(-1.0 + Float64(-1.0 / eps)))) / 2.0);
	elseif (x <= 8.8e-175)
		tmp = Float64(fma(Float64(1.0 + Float64(1.0 / eps)), 1.0, Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps + -1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.6e-224], N[(N[(N[(-1.0 / eps), $MachinePrecision] - N[(N[Exp[N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.8e-175], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0 + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.60000000000000005e-224

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in eps around inf 67.3%

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

      1. mul-1-neg 67.3%

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

      2. distribute-lft-neg-out 67.3%

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

      3. *-commutative 67.3%

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

   7. Simplified 67.3%

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

   8. Taylor expanded in eps around 0 30.6%

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

      1. *-commutative 30.6%

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

      2. sub-neg 30.6%

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

      3. distribute-lft-in 30.6%

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

      4. *-commutative 30.6%

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

      5. *-un-lft-identity 30.6%

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

      6. add-sqr-sqrt 30.6%

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

      7. sqrt-unprod 34.7%

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

      8. sqr-neg 34.7%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 30.6%

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

      11. add-sqr-sqrt 30.6%

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

      12. sqrt-unprod 29.7%

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

      13. sqr-neg 29.7%

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

      14. sqrt-unprod 0.0%

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

      15. add-sqr-sqrt 36.2%

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

   10. Applied egg-rr 36.2%

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

       1. *-rgt-identity 36.2%

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

       2. distribute-lft-in 36.2%

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

       3. sub-neg 36.2%

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

   12. Simplified 36.2%

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

   if -7.60000000000000005e-224 < x < 8.8e-175

   1. Initial program 58.0%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in x around 0 55.4%

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

   6. Taylor expanded in x around 0 54.5%

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

   if 8.8e-175 < x

   1. Initial program 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in eps around inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. distribute-lft-neg-out 55.5%

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

      3. *-commutative 55.5%

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

   7. Simplified 55.5%

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

   8. Taylor expanded in eps around 0 22.6%

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

   9. Taylor expanded in eps around inf 63.0%

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

       1. mul-1-neg 63.0%

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

       2. distribute-rgt-neg-in 63.0%

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

   11. Simplified 63.0%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\frac{-1}{\varepsilon} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, 1, 1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.08)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (/ (exp (* x (+ eps -1.0))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.08) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else {
		tmp = exp((x * (eps + -1.0))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.08) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else {
		tmp = Math.exp((x * (eps + -1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.08:
		tmp = (math.expm1(-x) / eps) / 2.0
	else:
		tmp = math.exp((x * (eps + -1.0))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.08)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps + -1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.08], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0800000000000001

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-neg-out 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 54.8%

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

   9. Taylor expanded in eps around 0 45.7%

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

       1. expm1-define 45.7%

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

       2. mul-1-neg 45.7%

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

   11. Simplified 45.7%

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

   if -1.0800000000000001 < x

   1. Initial program 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in eps around inf 54.2%

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

      1. mul-1-neg 54.2%

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

      2. distribute-lft-neg-out 54.2%

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

      3. *-commutative 54.2%

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

   7. Simplified 54.2%

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

   8. Taylor expanded in eps around 0 18.3%

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

   9. Taylor expanded in eps around inf 41.0%

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

       1. mul-1-neg 41.0%

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

       2. distribute-rgt-neg-in 41.0%

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

   11. Simplified 41.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 19.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -470:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 - \varepsilon\right) \cdot \left(x \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -470.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 0.014)
     0.5
     (/ (+ 1.0 (* (- 1.0 eps) (* x (+ -1.0 (/ -1.0 eps))))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -470.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 0.014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -470.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 0.014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -470.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 0.014:
		tmp = 0.5
	else:
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -470.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 0.014)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 - eps) * Float64(x * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -470.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.014], 0.5, N[(N[(1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(x * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -470

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-neg-out 100.0%

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

      3. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 54.8%

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

   9. Taylor expanded in eps around 0 45.7%

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

       1. expm1-define 45.7%

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

       2. mul-1-neg 45.7%

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

   11. Simplified 45.7%

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

   if -470 < x < 0.0140000000000000003

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in eps around inf 51.5%

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

      1. mul-1-neg 51.5%

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

      2. distribute-lft-neg-out 51.5%

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

      3. *-commutative 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in eps around 0 16.0%

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

   9. Taylor expanded in x around 0 15.2%

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

   if 0.0140000000000000003 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 60.2%

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

      1. mul-1-neg 60.2%

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

      2. distribute-lft-neg-out 60.2%

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

      3. *-commutative 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in eps around 0 23.5%

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

   9. Taylor expanded in x around 0 10.6%

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

       1. mul-1-neg 10.6%

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

       2. unsub-neg 10.6%

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

       3. associate-*r* 10.6%

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

   11. Simplified 10.6%

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

4. Final simplification 18.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -470:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 - \varepsilon\right) \cdot \left(x \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 13.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 - \varepsilon\right) \cdot \left(x \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.014)
   0.5
   (/ (+ 1.0 (* (- 1.0 eps) (* x (+ -1.0 (/ -1.0 eps))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.014d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 + ((1.0d0 - eps) * (x * ((-1.0d0) + ((-1.0d0) / eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.014:
		tmp = 0.5
	else:
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.014)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(1.0 - eps) * Float64(x * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.014)
		tmp = 0.5;
	else
		tmp = (1.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.014], 0.5, N[(N[(1.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(x * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0140000000000000003

   1. Initial program 60.6%

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

   3. Simplified 60.6%

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

   5. Taylor expanded in eps around inf 60.6%

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

      1. mul-1-neg 60.6%

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

      2. distribute-lft-neg-out 60.6%

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

      3. *-commutative 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in eps around 0 23.3%

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

   9. Taylor expanded in x around 0 12.9%

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

   if 0.0140000000000000003 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 60.2%

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

      1. mul-1-neg 60.2%

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

      2. distribute-lft-neg-out 60.2%

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

      3. *-commutative 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in eps around 0 23.5%

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

   9. Taylor expanded in x around 0 10.6%

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

       1. mul-1-neg 10.6%

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

       2. unsub-neg 10.6%

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

       3. associate-*r* 10.6%

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

   11. Simplified 10.6%

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

4. Final simplification 12.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 - \varepsilon\right) \cdot \left(x \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.0% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.5)

C

double code(double x, double eps) {
	return 0.5;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.5d0
end function

Java

public static double code(double x, double eps) {
	return 0.5;
}

Python

def code(x, eps):
	return 0.5

Julia

function code(x, eps)
	return 0.5
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5;
end

Wolfram

code[x_, eps_] := 0.5

Derivation

1. Initial program 71.2%

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

3. Simplified 71.2%

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

5. Taylor expanded in eps around inf 60.5%

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

   1. mul-1-neg 60.5%

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

   2. distribute-lft-neg-out 60.5%

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

   3. *-commutative 60.5%

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

7. Simplified 60.5%

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

8. Taylor expanded in eps around 0 23.3%

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

9. Taylor expanded in x around 0 10.3%

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

10. Final simplification 10.3%

    \[\leadsto 0.5 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))