NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.9%

Time: 15.6s

Alternatives: 11

Speedup: 2.1×

• 38.5% 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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(1 + x\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq 0.032:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 x) (exp (- x)))))
   (if (<= eps 0.032)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* eps x)) (exp (* x (- -1.0 eps)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (1.0 + x) * exp(-x);
	double tmp;
	if (eps <= 0.032) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) * exp(-x)
    if (eps <= 0.032d0) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((eps * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (1.0 + x) * Math.exp(-x);
	double tmp;
	if (eps <= 0.032) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (1.0 + x) * math.exp(-x)
	tmp = 0
	if eps <= 0.032:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((eps * x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(1.0 + x) * exp(Float64(-x)))
	tmp = 0.0
	if (eps <= 0.032)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (1.0 + x) * exp(-x);
	tmp = 0.0;
	if (eps <= 0.032)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 0.032], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.032000000000000001

   1. Initial program 61.4%

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

      1. div-sub 61.4%

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

      2. +-rgt-identity 61.4%

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

      3. div-sub 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in eps around 0 76.8%

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

      1. *-commutative 76.8%

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

      2. distribute-lft1-in 76.8%

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

      3. neg-mul-1 76.8%

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

      4. distribute-lft-out 76.8%

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

      5. mul-1-neg 76.8%

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

      6. *-commutative 76.8%

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

      7. distribute-lft1-in 76.8%

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

      8. neg-mul-1 76.8%

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

   6. Simplified 76.8%

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

   if 0.032000000000000001 < 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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. mul-1-neg 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. mul-1-neg 100.0%

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

      8. mul-1-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.032:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.55e-13)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ (exp (* eps x)) (exp (* x (- -1.0 eps)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1.55e-13) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.55d-13) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((eps * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.55e-13) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 1.55e-13:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((eps * x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 1.55e-13)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.55e-13)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((eps * x)) + exp((x * (-1.0 - eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 1.55e-13], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.55e-13

   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

      1. div-sub 61.6%

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

      2. +-rgt-identity 61.6%

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

      3. div-sub 61.6%

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

   3. Simplified 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 inf 99.1%

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

   5. Taylor expanded in eps around 0 79.8%

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

      1. cancel-sign-sub-inv 79.8%

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

      2. metadata-eval 79.8%

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

      3. distribute-rgt1-in 79.8%

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

      4. metadata-eval 79.8%

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

      5. neg-mul-1 79.8%

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

   7. Simplified 79.8%

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

   if 1.55e-13 < eps

   1. Initial program 99.2%

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

      1. div-sub 99.2%

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

      2. +-rgt-identity 99.2%

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

      3. div-sub 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in eps around inf 98.8%

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

   5. Taylor expanded in eps around -inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. mul-1-neg 98.8%

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

      3. sub-neg 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-lft-neg-in 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. mul-1-neg 98.8%

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

      8. mul-1-neg 98.8%

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

      9. *-commutative 98.8%

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

      10. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in eps around inf 98.8%

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

   9. Taylor expanded in eps around inf 98.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- -1.0 eps)))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp((x * (eps + (-1.0d0)))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

   1. div-sub 73.6%

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

   2. +-rgt-identity 73.6%

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

   3. div-sub 73.6%

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

3. Simplified 73.6%

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

4. Taylor expanded in eps around inf 99.0%

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

5. Taylor expanded in eps around -inf 99.0%

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

   1. associate-*r* 99.0%

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

   2. mul-1-neg 99.0%

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

   3. sub-neg 99.0%

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

   4. mul-1-neg 99.0%

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

   5. distribute-lft-neg-in 99.0%

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

   6. distribute-rgt-neg-in 99.0%

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

   7. mul-1-neg 99.0%

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

   8. mul-1-neg 99.0%

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

   9. *-commutative 99.0%

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

   10. mul-1-neg 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 4: 83.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ t_1 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 190000000000:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+181}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0))
        (t_1 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x -2e-250)
     (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
     (if (<= x 190000000000.0)
       (/ (- (exp (* eps x)) -1.0) 2.0)
       (if (<= x 6e+95)
         t_1
         (if (<= x 1.1e+132)
           t_0
           (if (<= x 1.3e+157) 0.0 (if (<= x 5.5e+181) t_0 t_1))))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	double t_1 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= -2e-250) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 190000000000.0) {
		tmp = (exp((eps * x)) - -1.0) / 2.0;
	} else if (x <= 6e+95) {
		tmp = t_1;
	} else if (x <= 1.1e+132) {
		tmp = t_0;
	} else if (x <= 1.3e+157) {
		tmp = 0.0;
	} else if (x <= 5.5e+181) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
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)))) - (-1.0d0)) / 2.0d0
    t_1 = (2.0d0 * exp(-x)) / 2.0d0
    if (x <= (-2d-250)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 190000000000.0d0) then
        tmp = (exp((eps * x)) - (-1.0d0)) / 2.0d0
    else if (x <= 6d+95) then
        tmp = t_1
    else if (x <= 1.1d+132) then
        tmp = t_0
    else if (x <= 1.3d+157) then
        tmp = 0.0d0
    else if (x <= 5.5d+181) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	double t_1 = (2.0 * Math.exp(-x)) / 2.0;
	double tmp;
	if (x <= -2e-250) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 190000000000.0) {
		tmp = (Math.exp((eps * x)) - -1.0) / 2.0;
	} else if (x <= 6e+95) {
		tmp = t_1;
	} else if (x <= 1.1e+132) {
		tmp = t_0;
	} else if (x <= 1.3e+157) {
		tmp = 0.0;
	} else if (x <= 5.5e+181) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	t_1 = (2.0 * math.exp(-x)) / 2.0
	tmp = 0
	if x <= -2e-250:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 190000000000.0:
		tmp = (math.exp((eps * x)) - -1.0) / 2.0
	elif x <= 6e+95:
		tmp = t_1
	elif x <= 1.1e+132:
		tmp = t_0
	elif x <= 1.3e+157:
		tmp = 0.0
	elif x <= 5.5e+181:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0)
	t_1 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= -2e-250)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 190000000000.0)
		tmp = Float64(Float64(exp(Float64(eps * x)) - -1.0) / 2.0);
	elseif (x <= 6e+95)
		tmp = t_1;
	elseif (x <= 1.1e+132)
		tmp = t_0;
	elseif (x <= 1.3e+157)
		tmp = 0.0;
	elseif (x <= 5.5e+181)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	t_1 = (2.0 * exp(-x)) / 2.0;
	tmp = 0.0;
	if (x <= -2e-250)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 190000000000.0)
		tmp = (exp((eps * x)) - -1.0) / 2.0;
	elseif (x <= 6e+95)
		tmp = t_1;
	elseif (x <= 1.1e+132)
		tmp = t_0;
	elseif (x <= 1.3e+157)
		tmp = 0.0;
	elseif (x <= 5.5e+181)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2e-250], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 190000000000.0], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+95], t$95$1, If[LessEqual[x, 1.1e+132], t$95$0, If[LessEqual[x, 1.3e+157], 0.0, If[LessEqual[x, 5.5e+181], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.0000000000000001e-250

   1. Initial program 74.4%

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

      1. div-sub 74.4%

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

      2. +-rgt-identity 74.4%

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

      3. div-sub 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x around 0 54.8%

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

   5. Taylor expanded in eps around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. exp-prod 79.3%

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

      3. +-commutative 79.3%

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

      4. remove-double-neg 79.3%

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

      5. mul-1-neg 79.3%

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

      6. sub-neg 79.3%

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

      7. exp-prod 79.3%

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

      8. associate-*r* 79.3%

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

      9. cancel-sign-sub-inv 79.3%

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

      10. metadata-eval 79.3%

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

      11. *-lft-identity 79.3%

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

      12. distribute-lft-in 79.3%

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

      13. metadata-eval 79.3%

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

      14. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

   if -2.0000000000000001e-250 < x < 1.9e11

   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

      1. div-sub 53.1%

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

      2. +-rgt-identity 53.1%

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

      3. div-sub 53.1%

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

   3. Simplified 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 eps around inf 98.8%

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

   5. Taylor expanded in eps around -inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. mul-1-neg 98.8%

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

      3. sub-neg 98.8%

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

      4. mul-1-neg 98.8%

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

      5. distribute-lft-neg-in 98.8%

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

      6. distribute-rgt-neg-in 98.8%

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

      7. mul-1-neg 98.8%

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

      8. mul-1-neg 98.8%

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

      9. *-commutative 98.8%

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

      10. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in eps around inf 97.7%

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

   9. Taylor expanded in x around 0 83.3%

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

   if 1.9e11 < x < 5.99999999999999982e95 or 5.49999999999999991e181 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 76.5%

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

      1. cancel-sign-sub-inv 76.5%

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

      2. metadata-eval 76.5%

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

      3. distribute-rgt1-in 76.5%

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

      4. metadata-eval 76.5%

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

      5. neg-mul-1 76.5%

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

   7. Simplified 76.5%

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

   if 5.99999999999999982e95 < x < 1.09999999999999994e132 or 1.30000000000000005e157 < x < 5.49999999999999991e181

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. mul-1-neg 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. mul-1-neg 100.0%

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

      8. mul-1-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 35.4%

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

   if 1.09999999999999994e132 < x < 1.30000000000000005e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. div-sub 100.0%

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

      2. rec-exp 100.0%

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

      3. neg-mul-1 100.0%

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

      4. +-inverses 100.0%

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

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-250}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 190000000000:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+95}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+181}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 5: 83.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{e^{\varepsilon \cdot x} - -1}{2}\\ t_1 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 15500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+179}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (- (exp (* eps x)) -1.0) 2.0))
        (t_1 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x -5e-250)
     (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
     (if (<= x 15500000000000.0)
       t_0
       (if (<= x 1.3e+96)
         t_1
         (if (<= x 4.4e+132)
           t_0
           (if (<= x 2.5e+157) 0.0 (if (<= x 7e+179) t_0 t_1))))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (exp((eps * x)) - -1.0) / 2.0;
	double t_1 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= -5e-250) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 15500000000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e+96) {
		tmp = t_1;
	} else if (x <= 4.4e+132) {
		tmp = t_0;
	} else if (x <= 2.5e+157) {
		tmp = 0.0;
	} else if (x <= 7e+179) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
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((eps * x)) - (-1.0d0)) / 2.0d0
    t_1 = (2.0d0 * exp(-x)) / 2.0d0
    if (x <= (-5d-250)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 15500000000000.0d0) then
        tmp = t_0
    else if (x <= 1.3d+96) then
        tmp = t_1
    else if (x <= 4.4d+132) then
        tmp = t_0
    else if (x <= 2.5d+157) then
        tmp = 0.0d0
    else if (x <= 7d+179) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (Math.exp((eps * x)) - -1.0) / 2.0;
	double t_1 = (2.0 * Math.exp(-x)) / 2.0;
	double tmp;
	if (x <= -5e-250) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 15500000000000.0) {
		tmp = t_0;
	} else if (x <= 1.3e+96) {
		tmp = t_1;
	} else if (x <= 4.4e+132) {
		tmp = t_0;
	} else if (x <= 2.5e+157) {
		tmp = 0.0;
	} else if (x <= 7e+179) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (math.exp((eps * x)) - -1.0) / 2.0
	t_1 = (2.0 * math.exp(-x)) / 2.0
	tmp = 0
	if x <= -5e-250:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 15500000000000.0:
		tmp = t_0
	elif x <= 1.3e+96:
		tmp = t_1
	elif x <= 4.4e+132:
		tmp = t_0
	elif x <= 2.5e+157:
		tmp = 0.0
	elif x <= 7e+179:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(exp(Float64(eps * x)) - -1.0) / 2.0)
	t_1 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= -5e-250)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 15500000000000.0)
		tmp = t_0;
	elseif (x <= 1.3e+96)
		tmp = t_1;
	elseif (x <= 4.4e+132)
		tmp = t_0;
	elseif (x <= 2.5e+157)
		tmp = 0.0;
	elseif (x <= 7e+179)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (exp((eps * x)) - -1.0) / 2.0;
	t_1 = (2.0 * exp(-x)) / 2.0;
	tmp = 0.0;
	if (x <= -5e-250)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 15500000000000.0)
		tmp = t_0;
	elseif (x <= 1.3e+96)
		tmp = t_1;
	elseif (x <= 4.4e+132)
		tmp = t_0;
	elseif (x <= 2.5e+157)
		tmp = 0.0;
	elseif (x <= 7e+179)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -5e-250], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 15500000000000.0], t$95$0, If[LessEqual[x, 1.3e+96], t$95$1, If[LessEqual[x, 4.4e+132], t$95$0, If[LessEqual[x, 2.5e+157], 0.0, If[LessEqual[x, 7e+179], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.00000000000000027e-250

   1. Initial program 74.4%

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

      1. div-sub 74.4%

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

      2. +-rgt-identity 74.4%

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

      3. div-sub 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x around 0 54.8%

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

   5. Taylor expanded in eps around inf 79.3%

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

      1. mul-1-neg 79.3%

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

      2. exp-prod 79.3%

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

      3. +-commutative 79.3%

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

      4. remove-double-neg 79.3%

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

      5. mul-1-neg 79.3%

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

      6. sub-neg 79.3%

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

      7. exp-prod 79.3%

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

      8. associate-*r* 79.3%

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

      9. cancel-sign-sub-inv 79.3%

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

      10. metadata-eval 79.3%

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

      11. *-lft-identity 79.3%

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

      12. distribute-lft-in 79.3%

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

      13. metadata-eval 79.3%

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

      14. mul-1-neg 79.3%

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

   7. Simplified 79.3%

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

   if -5.00000000000000027e-250 < x < 1.55e13 or 1.3e96 < x < 4.39999999999999977e132 or 2.49999999999999988e157 < x < 7.0000000000000003e179

   1. Initial program 59.1%

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

      1. div-sub 59.1%

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

      2. +-rgt-identity 59.1%

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

      3. div-sub 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in eps around -inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. mul-1-neg 98.9%

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

      3. sub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-lft-neg-in 98.9%

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

      6. distribute-rgt-neg-in 98.9%

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

      7. mul-1-neg 98.9%

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

      8. mul-1-neg 98.9%

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

      9. *-commutative 98.9%

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

      10. mul-1-neg 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in eps around inf 97.0%

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

   9. Taylor expanded in x around 0 76.8%

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

   if 1.55e13 < x < 1.3e96 or 7.0000000000000003e179 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 75.4%

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

      1. cancel-sign-sub-inv 75.4%

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

      2. metadata-eval 75.4%

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

      3. distribute-rgt1-in 75.4%

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

      4. metadata-eval 75.4%

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

      5. neg-mul-1 75.4%

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

   7. Simplified 75.4%

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

   if 4.39999999999999977e132 < x < 2.49999999999999988e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. div-sub 100.0%

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

      2. rec-exp 100.0%

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

      3. neg-mul-1 100.0%

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

      4. +-inverses 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-250}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 15500000000000:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+96}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+179}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 6: 77.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.25e+74)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (- (exp (* eps x)) -1.0) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1.25e+74) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((eps * x)) - -1.0) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.25d+74) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((eps * x)) - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.25e+74) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) - -1.0) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 1.25e+74:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((eps * x)) - -1.0) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 1.25e+74)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.25e+74)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((eps * x)) - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 1.25e+74], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.24999999999999991e74

   1. Initial program 66.1%

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

      1. div-sub 66.1%

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

      2. +-rgt-identity 66.1%

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

      3. div-sub 66.1%

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

   3. Simplified 66.1%

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

   4. Taylor expanded in eps around inf 98.7%

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

   5. Taylor expanded in eps around 0 79.9%

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

      1. cancel-sign-sub-inv 79.9%

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

      2. metadata-eval 79.9%

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

      3. distribute-rgt1-in 79.9%

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

      4. metadata-eval 79.9%

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

      5. neg-mul-1 79.9%

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

   7. Simplified 79.9%

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

   if 1.24999999999999991e74 < 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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. mul-1-neg 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. mul-1-neg 100.0%

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

      8. mul-1-neg 100.0%

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

      9. *-commutative 100.0%

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

      10. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

   9. Taylor expanded in x around 0 52.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} - -1}{2}\\ \end{array} \]

Alternative 7: 70.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \frac{2 \cdot e^{-x}}{2} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (/ (* 2.0 (exp (- x))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 * exp(-x)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.6%

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

   1. div-sub 73.6%

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

   2. +-rgt-identity 73.6%

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

   3. div-sub 73.6%

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

3. Simplified 73.6%

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

4. Taylor expanded in eps around inf 99.0%

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

5. Taylor expanded in eps around 0 73.6%

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

   1. cancel-sign-sub-inv 73.6%

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

   2. metadata-eval 73.6%

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

   3. distribute-rgt1-in 73.6%

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

   4. metadata-eval 73.6%

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

   5. neg-mul-1 73.6%

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

7. Simplified 73.6%

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

8. Final simplification 73.6%

   \[\leadsto \frac{2 \cdot e^{-x}}{2} \]

Alternative 8: 65.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := -1 + \frac{1}{\varepsilon}\\ t_1 := \left(x \cdot \left(-1 - \varepsilon\right)\right) \cdot t_0\\ \mathbf{if}\;x \leq -6 \cdot 10^{+123}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq -3.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{4 + \left(\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot t_0\right) \cdot t_1}{2 + t_1}}{2}\\ \mathbf{elif}\;x \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 1.0 eps))) (t_1 (* (* x (- -1.0 eps)) t_0)))
   (if (<= x -6e+123)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x -3.55e-15)
       (/ (/ (+ 4.0 (* (* (* x (+ 1.0 eps)) t_0) t_1)) (+ 2.0 t_1)) 2.0)
       (if (<= x 10500.0) 1.0 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = -1.0 + (1.0 / eps);
	double t_1 = (x * (-1.0 - eps)) * t_0;
	double tmp;
	if (x <= -6e+123) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= -3.55e-15) {
		tmp = ((4.0 + (((x * (1.0 + eps)) * t_0) * t_1)) / (2.0 + t_1)) / 2.0;
	} else if (x <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + (1.0d0 / eps)
    t_1 = (x * ((-1.0d0) - eps)) * t_0
    if (x <= (-6d+123)) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= (-3.55d-15)) then
        tmp = ((4.0d0 + (((x * (1.0d0 + eps)) * t_0) * t_1)) / (2.0d0 + t_1)) / 2.0d0
    else if (x <= 10500.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = -1.0 + (1.0 / eps);
	double t_1 = (x * (-1.0 - eps)) * t_0;
	double tmp;
	if (x <= -6e+123) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= -3.55e-15) {
		tmp = ((4.0 + (((x * (1.0 + eps)) * t_0) * t_1)) / (2.0 + t_1)) / 2.0;
	} else if (x <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = -1.0 + (1.0 / eps)
	t_1 = (x * (-1.0 - eps)) * t_0
	tmp = 0
	if x <= -6e+123:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= -3.55e-15:
		tmp = ((4.0 + (((x * (1.0 + eps)) * t_0) * t_1)) / (2.0 + t_1)) / 2.0
	elif x <= 10500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(-1.0 + Float64(1.0 / eps))
	t_1 = Float64(Float64(x * Float64(-1.0 - eps)) * t_0)
	tmp = 0.0
	if (x <= -6e+123)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= -3.55e-15)
		tmp = Float64(Float64(Float64(4.0 + Float64(Float64(Float64(x * Float64(1.0 + eps)) * t_0) * t_1)) / Float64(2.0 + t_1)) / 2.0);
	elseif (x <= 10500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = -1.0 + (1.0 / eps);
	t_1 = (x * (-1.0 - eps)) * t_0;
	tmp = 0.0;
	if (x <= -6e+123)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= -3.55e-15)
		tmp = ((4.0 + (((x * (1.0 + eps)) * t_0) * t_1)) / (2.0 + t_1)) / 2.0;
	elseif (x <= 10500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x, -6e+123], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -3.55e-15], N[(N[(N[(4.0 + N[(N[(N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 10500.0], 1.0, 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.00000000000000016e123

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 84.7%

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

   5. Taylor expanded in x around 0 69.4%

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

   6. Taylor expanded in eps around inf 69.4%

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

      1. associate-+r+ 69.4%

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

      2. metadata-eval 69.4%

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

      3. associate-*r* 69.4%

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

      4. neg-mul-1 69.4%

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

      5. *-commutative 69.4%

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

      6. associate-+r+ 69.4%

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

      7. +-commutative 69.4%

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

      8. distribute-lft1-in 69.4%

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

      9. metadata-eval 69.4%

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

      10. mul0-lft 69.4%

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

   8. Simplified 69.4%

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

   if -6.00000000000000016e123 < x < -3.5500000000000001e-15

   1. Initial program 92.0%

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

      1. div-sub 92.0%

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

      2. +-rgt-identity 92.0%

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

      3. div-sub 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in x around 0 61.5%

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

   5. Taylor expanded in x around 0 9.0%

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

      1. flip-+ 26.9%

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

      2. metadata-eval 26.9%

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

      3. sub-neg 26.9%

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

      4. metadata-eval 26.9%

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

      5. *-commutative 26.9%

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

      6. *-commutative 26.9%

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

      7. +-commutative 26.9%

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

      8. sub-neg 26.9%

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

      9. metadata-eval 26.9%

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

      10. *-commutative 26.9%

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

      11. *-commutative 26.9%

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

      12. +-commutative 26.9%

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

   7. Applied egg-rr 26.9%

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

   if -3.5500000000000001e-15 < x < 10500

   1. Initial program 55.8%

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

      1. div-sub 55.8%

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

      2. +-rgt-identity 55.8%

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

      3. div-sub 55.8%

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

   3. Simplified 55.8%

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

   4. Taylor expanded in x around 0 74.2%

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

   if 10500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.8%

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

      1. div-sub 63.8%

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

      2. rec-exp 63.8%

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

      3. neg-mul-1 63.8%

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

      4. +-inverses 63.8%

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

   6. Simplified 63.8%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+123}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq -3.55 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{4 + \left(\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\left(x \cdot \left(-1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2 + \left(x \cdot \left(-1 - \varepsilon\right)\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}}{2}\\ \mathbf{elif}\;x \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 63.4% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 235.0) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 235.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 235.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 235.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 235.0:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 235.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 235.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 235.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 235

   1. Initial program 63.9%

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

      1. div-sub 63.9%

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

      2. +-rgt-identity 63.9%

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

      3. div-sub 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in x around 0 49.9%

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

   5. Taylor expanded in x around 0 47.4%

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

   6. Taylor expanded in eps around inf 67.0%

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

      1. associate-+r+ 67.0%

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

      2. metadata-eval 67.0%

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

      3. associate-*r* 67.0%

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

      4. neg-mul-1 67.0%

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

      5. *-commutative 67.0%

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

      6. associate-+r+ 67.0%

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

      7. +-commutative 67.0%

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

      8. distribute-lft1-in 67.0%

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

      9. metadata-eval 67.0%

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

      10. mul0-lft 67.0%

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

   8. Simplified 67.0%

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

   if 235 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 62.9%

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

      1. div-sub 62.9%

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

      2. rec-exp 62.9%

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

      3. neg-mul-1 62.9%

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

      4. +-inverses 62.9%

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

   6. Simplified 62.9%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 56.8% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 10500.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 10500.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 10500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 10500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 10500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 10500.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 10500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 10500

   1. Initial program 64.1%

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

      1. div-sub 64.1%

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

      2. +-rgt-identity 64.1%

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

      3. div-sub 64.1%

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

   3. Simplified 64.1%

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

   4. Taylor expanded in x around 0 59.9%

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

   if 10500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.8%

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

      1. div-sub 63.8%

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

      2. rec-exp 63.8%

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

      3. neg-mul-1 63.8%

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

      4. +-inverses 63.8%

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

   6. Simplified 63.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 16.4% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 73.6%

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

3. Simplified 67.0%

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

4. Taylor expanded in eps around 0 18.6%

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

   1. div-sub 18.6%

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

   2. rec-exp 18.6%

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

   3. neg-mul-1 18.6%

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

   4. +-inverses 18.8%

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

6. Simplified 18.8%

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

7. Final simplification 18.8%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023257 
(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))