NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.9% → 100.0%

Time: 11.9s

Alternatives: 12

Speedup: 2.0×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;eps_m \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps_m}\right) \cdot e^{eps_m \cdot x - x} + e^{x \cdot \left(-1 - eps_m\right)} \cdot \left(1 + \frac{-1}{eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps_m 2e-16)
     (/ (+ t_0 t_0) 2.0)
     (/
      (+
       (* (- 1.0 (/ -1.0 eps_m)) (exp (- (* eps_m x) x)))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps_m <= 2e-16) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 - (-1.0 / eps_m)) * exp(((eps_m * x) - x))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps_m <= 2d-16) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (((1.0d0 - ((-1.0d0) / eps_m)) * exp(((eps_m * x) - x))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 2e-16) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 - (-1.0 / eps_m)) * Math.exp(((eps_m * x) - x))) + (Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 2e-16:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (((1.0 - (-1.0 / eps_m)) * math.exp(((eps_m * x) - x))) + (math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 2e-16)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(-1.0 / eps_m)) * exp(Float64(Float64(eps_m * x) - x))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 2e-16)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (((1.0 - (-1.0 / eps_m)) * exp(((eps_m * x) - x))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-16], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(eps$95$m * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2e-16

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in eps around 0 70.2%

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

   6. Simplified 71.3%

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

   if 2e-16 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;eps_m \leq 5 \cdot 10^{-17}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps_m}\right) \cdot e^{x \cdot \left(eps_m + -1\right)} + e^{x \cdot \left(-1 - eps_m\right)} \cdot \left(1 + \frac{-1}{eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps_m 5e-17)
     (/ (+ t_0 t_0) 2.0)
     (/
      (+
       (* (- 1.0 (/ -1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps_m <= 5e-17) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 - (-1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps_m <= 5d-17) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (((1.0d0 - ((-1.0d0) / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 5e-17) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 - (-1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) + (Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 5e-17:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (((1.0 - (-1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) + (math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 5e-17)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(-1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 5e-17)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (((1.0 - (-1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 5e-17], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.9999999999999999e-17

   1. Initial program 61.6%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in eps around 0 70.2%

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

   6. Simplified 71.3%

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

   if 4.9999999999999999e-17 < 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. Recombined 2 regimes into one program.

4. Final simplification 79.6%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;eps_m \leq 0.0054:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{eps_m \cdot x} + e^{x \cdot \left(-1 - eps_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps_m 0.0054)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* eps_m x)) (exp (* x (- -1.0 eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps_m <= 0.0054) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((eps_m * x)) + exp((x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps_m <= 0.0054d0) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((eps_m * x)) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 0.0054) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((eps_m * x)) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 0.0054:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((eps_m * x)) + math.exp((x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 0.0054)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps_m * x)) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.0054)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((eps_m * x)) + exp((x * (-1.0 - eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.0054], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0054000000000000003

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in eps around 0 70.7%

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

   6. Simplified 71.7%

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

   if 0.0054000000000000003 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 99.4%

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

   5. Taylor expanded in eps around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 99.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0054:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{eps_m \cdot x} + e^{eps_m \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.1e+154)
   (/ (+ (exp (* eps_m x)) (exp (* eps_m (- x)))) 2.0)
   (/ (/ 0.0 eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.1e+154) {
		tmp = (exp((eps_m * x)) + exp((eps_m * -x))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.1d+154) then
        tmp = (exp((eps_m * x)) + exp((eps_m * -x))) / 2.0d0
    else
        tmp = (0.0d0 / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.1e+154) {
		tmp = (Math.exp((eps_m * x)) + Math.exp((eps_m * -x))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.1e+154:
		tmp = (math.exp((eps_m * x)) + math.exp((eps_m * -x))) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.1e+154)
		tmp = Float64(Float64(exp(Float64(eps_m * x)) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.1e+154)
		tmp = (exp((eps_m * x)) + exp((eps_m * -x))) / 2.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.1e+154], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001e154

   1. Initial program 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in eps around inf 97.9%

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

   5. Taylor expanded in eps around inf 90.7%

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

      1. *-commutative 90.7%

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

   7. Simplified 90.7%

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

   8. Taylor expanded in x around inf 90.7%

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

      1. cancel-sign-sub-inv 90.7%

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

      2. metadata-eval 90.7%

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

      3. *-lft-identity 90.7%

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

      4. +-commutative 90.7%

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

      5. mul-1-neg 90.7%

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

      6. distribute-rgt-neg-in 90.7%

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

      7. mul-1-neg 90.7%

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

      8. distribute-rgt-neg-in 90.7%

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

      9. sub-neg 90.7%

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

      10. mul-1-neg 90.7%

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

      11. distribute-neg-in 90.7%

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

      12. metadata-eval 90.7%

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

      13. mul-1-neg 90.7%

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

      14. remove-double-neg 90.7%

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

   10. Simplified 90.7%

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

   11. Taylor expanded in eps around inf 90.7%

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

       1. *-commutative 90.7%

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

   13. Simplified 90.7%

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

   if 1.1000000000000001e154 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 75.4%

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

   5. Taylor expanded in x around 0 75.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 5: 92.5% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* eps_m (- x))) (exp (* x (+ eps_m -1.0)))) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

4. Taylor expanded in eps around inf 98.2%

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

5. Taylor expanded in eps around inf 87.2%

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

   1. *-commutative 87.2%

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

7. Simplified 87.2%

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

8. Taylor expanded in x around inf 87.2%

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

   1. cancel-sign-sub-inv 87.2%

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

   2. metadata-eval 87.2%

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

   3. *-lft-identity 87.2%

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

   4. +-commutative 87.2%

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

   5. mul-1-neg 87.2%

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

   6. distribute-rgt-neg-in 87.2%

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

   7. mul-1-neg 87.2%

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

   8. distribute-rgt-neg-in 87.2%

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

   9. sub-neg 87.2%

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

   10. mul-1-neg 87.2%

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

   11. distribute-neg-in 87.2%

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

   12. metadata-eval 87.2%

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

   13. mul-1-neg 87.2%

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

   14. remove-double-neg 87.2%

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

10. Simplified 87.2%

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

11. Final simplification 87.2%

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

Alternative 6: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -85000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -85000000.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 1.1e+154)
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     (/ (/ 0.0 eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -85000000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 1.1e+154) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -85000000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 1.1e+154) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -85000000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 1.1e+154:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -85000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 1.1e+154)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -85000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.1e+154], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 66.9%

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

   5. Taylor expanded in eps around 0 34.2%

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

      1. expm1-def 34.2%

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

      2. mul-1-neg 34.2%

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

   7. Simplified 34.2%

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

   if -8.5e7 < x < 1.1000000000000001e154

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in x around 0 42.3%

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

   5. Taylor expanded in eps around inf 75.9%

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

      1. mul-1-neg 75.9%

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

      2. distribute-rgt-neg-in 75.9%

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

      3. sub-neg 75.9%

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

      4. mul-1-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. metadata-eval 75.9%

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

      7. mul-1-neg 75.9%

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

      8. remove-double-neg 75.9%

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

   7. Simplified 75.9%

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

   if 1.1000000000000001e154 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 75.4%

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

   5. Taylor expanded in x around 0 75.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -85000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 7: 84.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-257)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 3.3e+152)
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     (/ (/ 0.0 eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-257) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 3.3e+152) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-4d-257)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 3.3d+152) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = (0.0d0 / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-257) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 3.3e+152) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4e-257:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 3.3e+152:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-257)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 3.3e+152)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -4e-257)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 3.3e+152)
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e-257], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.3e+152], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999999e-257

   1. Initial program 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in eps around inf 62.2%

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

   6. Taylor expanded in eps around -inf 62.2%

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

      1. sub-neg 62.2%

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

      2. mul-1-neg 62.2%

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

      3. remove-double-neg 62.2%

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

      4. mul-1-neg 62.2%

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

      5. distribute-rgt-neg-in 62.2%

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

      6. sub-neg 62.2%

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

      7. neg-mul-1 62.2%

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

      8. remove-double-neg 62.2%

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

      9. distribute-neg-in 62.2%

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

      10. metadata-eval 62.2%

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

      11. sub-neg 62.2%

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

   8. Simplified 62.2%

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

   if -3.9999999999999999e-257 < x < 3.3000000000000001e152

   1. Initial program 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in x around 0 42.5%

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

   5. Taylor expanded in eps around inf 73.0%

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

      1. mul-1-neg 73.0%

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

      2. distribute-rgt-neg-in 73.0%

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

      3. sub-neg 73.0%

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

      4. mul-1-neg 73.0%

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

      5. distribute-neg-in 73.0%

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

      6. metadata-eval 73.0%

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

      7. mul-1-neg 73.0%

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

      8. remove-double-neg 73.0%

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

   7. Simplified 73.0%

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

   if 3.3000000000000001e152 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 75.4%

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

   5. Taylor expanded in x around 0 75.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+152}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 8: 70.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -85000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 60000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -85000000.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 60000000000.0)
     1.0
     (if (<= x 5e+153) (/ (/ (expm1 x) eps_m) 2.0) (/ (/ 0.0 eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -85000000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 60000000000.0) {
		tmp = 1.0;
	} else if (x <= 5e+153) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -85000000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 60000000000.0) {
		tmp = 1.0;
	} else if (x <= 5e+153) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -85000000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 60000000000.0:
		tmp = 1.0
	elif x <= 5e+153:
		tmp = (math.expm1(x) / eps_m) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -85000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 60000000000.0)
		tmp = 1.0;
	elseif (x <= 5e+153)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -85000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 60000000000.0], 1.0, If[LessEqual[x, 5e+153], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.5e7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 66.9%

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

   5. Taylor expanded in eps around 0 34.2%

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

      1. expm1-def 34.2%

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

      2. mul-1-neg 34.2%

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

   7. Simplified 34.2%

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

   if -8.5e7 < x < 6e10

   1. Initial program 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in x around 0 73.2%

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

   if 6e10 < x < 5.00000000000000018e153

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 40.0%

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

   5. Taylor expanded in x around 0 1.7%

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

      1. sub-neg 1.7%

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

      2. neg-mul-1 1.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 42.1%

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

      5. sqr-neg 42.1%

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

      6. sqrt-unprod 42.1%

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

      7. add-sqr-sqrt 42.1%

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

      8. metadata-eval 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. rem-square-sqrt 42.1%

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

      2. fma-udef 42.1%

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

      3. metadata-eval 42.1%

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

      4. fma-neg 42.1%

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

      5. rem-square-sqrt 42.1%

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

      6. expm1-def 42.1%

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

   9. Simplified 42.1%

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

   if 5.00000000000000018e153 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 75.4%

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

   5. Taylor expanded in x around 0 75.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -85000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 60000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 9: 64.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps_m\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9e-8)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 2e+150) (/ (/ (expm1 x) eps_m) 2.0) (/ (/ 0.0 eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e-8) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 2e+150) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e-8) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 2e+150) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9e-8:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 2e+150:
		tmp = (math.expm1(x) / eps_m) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9e-8)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 2e+150)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9e-8], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+150], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.99999999999999986e-8

   1. Initial program 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 37.1%

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

   5. Taylor expanded in eps around inf 72.9%

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

   6. Taylor expanded in x around 0 62.8%

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

      1. mul-1-neg 62.8%

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

      2. +-commutative 62.8%

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

   8. Simplified 62.8%

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

   if 8.99999999999999986e-8 < x < 1.99999999999999996e150

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 40.9%

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

   5. Taylor expanded in x around 0 1.7%

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

      1. sub-neg 1.7%

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

      2. neg-mul-1 1.7%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 38.5%

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

      5. sqr-neg 38.5%

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

      6. sqrt-unprod 38.5%

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

      7. add-sqr-sqrt 38.5%

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

      8. metadata-eval 38.5%

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

   7. Applied egg-rr 38.5%

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

      1. rem-square-sqrt 38.5%

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

      2. fma-udef 38.5%

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

      3. metadata-eval 38.5%

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

      4. fma-neg 38.5%

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

      5. rem-square-sqrt 38.5%

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

      6. expm1-def 38.5%

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

   9. Simplified 38.5%

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

   if 1.99999999999999996e150 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 75.4%

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

   5. Taylor expanded in x around 0 75.4%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 10: 64.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9e-8) (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0) (/ (/ 0.0 eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e-8) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e-8) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9e-8:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9e-8)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	else
		tmp = Float64(Float64(0.0 / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 9e-8)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9e-8], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.99999999999999986e-8

   1. Initial program 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 37.1%

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

   5. Taylor expanded in eps around inf 72.9%

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

   6. Taylor expanded in x around 0 62.8%

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

      1. mul-1-neg 62.8%

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

      2. +-commutative 62.8%

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

   8. Simplified 62.8%

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

   if 8.99999999999999986e-8 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 54.1%

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

   5. Taylor expanded in x around 0 54.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]

Alternative 11: 58.1% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0) (/ (- 2.0 x) 2.0) (/ (/ 0.0 eps_m) 2.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in eps around inf 97.5%

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

   5. Taylor expanded in eps around inf 97.5%

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

      1. mul-1-neg 97.5%

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

      2. *-commutative 97.5%

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

      3. distribute-lft-neg-in 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in x around 0 59.7%

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

      1. mul-1-neg 59.7%

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

      2. unsub-neg 59.7%

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

   10. Simplified 59.7%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.6%

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

   5. Taylor expanded in x around 0 55.6%

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

4. Final simplification 58.6%

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

Alternative 12: 44.4% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

4. Taylor expanded in x around 0 43.9%

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

5. Final simplification 43.9%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023315 
(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))