NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.2%

Time: 14.1s

Alternatives: 15

Speedup: 2.0×

• 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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.4)
   (/ (+ (exp (* x eps)) (exp (* x (- eps)))) 2.0)
   (/ (+ (exp (- (* x eps) x)) (exp (- x))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.4) {
		tmp = (exp((x * eps)) + exp((x * -eps))) / 2.0;
	} else {
		tmp = (exp(((x * eps) - x)) + exp(-x)) / 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 (x <= 2.4d0) then
        tmp = (exp((x * eps)) + exp((x * -eps))) / 2.0d0
    else
        tmp = (exp(((x * eps) - x)) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.4], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 63.5%

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

      1. div-sub 63.5%

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

      2. +-rgt-identity 63.5%

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

      3. div-sub 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in eps around inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. associate-*r* 98.5%

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

      5. +-commutative 98.5%

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

      6. associate-*r* 98.5%

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

      7. mul-1-neg 98.5%

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

      8. *-commutative 98.5%

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

      9. distribute-rgt-neg-in 98.5%

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

      10. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in eps around inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   9. Simplified 98.5%

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

   10. Taylor expanded in x around inf 98.5%

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

       1. mul-1-neg 98.5%

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

   12. Simplified 98.5%

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

   13. Taylor expanded in eps around inf 99.3%

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

   if 2.39999999999999991 < x

   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 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. Step-by-step derivation

      1. mul-1-neg 99.0%

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

      2. *-commutative 99.0%

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

      3. mul-1-neg 99.0%

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

      4. associate-*r* 99.0%

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

      5. +-commutative 99.0%

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

      6. associate-*r* 99.0%

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

      7. mul-1-neg 99.0%

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

      8. *-commutative 99.0%

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

      9. distribute-rgt-neg-in 99.0%

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

      10. +-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in eps around 0 69.3%

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

      1. neg-mul-1 69.3%

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

   9. Simplified 69.3%

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

   10. Taylor expanded in x around inf 69.3%

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

4. Final simplification 89.2%

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

Alternative 2: 98.8% 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 75.5%

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

   1. div-sub 75.5%

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

   2. +-rgt-identity 75.5%

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

   3. div-sub 75.5%

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

3. Simplified 75.5%

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

4. Taylor expanded in 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. Step-by-step derivation

   1. mul-1-neg 98.7%

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

   2. *-commutative 98.7%

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

   3. mul-1-neg 98.7%

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

   4. associate-*r* 98.7%

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

   5. +-commutative 98.7%

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

   6. associate-*r* 98.7%

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

   7. mul-1-neg 98.7%

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

   8. *-commutative 98.7%

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

   9. distribute-rgt-neg-in 98.7%

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

   10. +-commutative 98.7%

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

6. Simplified 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.4e-7)
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (+ (exp (* x eps)) (exp (* x (- eps)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 1.4e-7) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * eps)) + exp((x * -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 <= 1.4d-7) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (exp((x * eps)) + exp((x * -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 <= 1.4e-7) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (Math.exp((x * eps)) + Math.exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 1.4e-7:
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (math.exp((x * eps)) + math.exp((x * -eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 1.4e-7)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.4e-7)
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (exp((x * eps)) + exp((x * -eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 1.4e-7], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.4000000000000001e-7

   1. Initial program 66.5%

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

      1. div-sub 66.5%

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

      2. +-rgt-identity 66.5%

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

      3. div-sub 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in eps around inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. *-commutative 98.2%

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

      3. mul-1-neg 98.2%

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

      4. associate-*r* 98.2%

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

      5. +-commutative 98.2%

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

      6. associate-*r* 98.2%

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

      7. mul-1-neg 98.2%

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

      8. *-commutative 98.2%

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

      9. distribute-rgt-neg-in 98.2%

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

      10. +-commutative 98.2%

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

   6. Simplified 98.2%

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

   7. Taylor expanded in eps around 0 79.5%

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

      1. neg-mul-1 79.5%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in eps around 0 75.2%

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

       1. exp-neg 75.2%

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

       2. associate-*r/ 75.2%

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

       3. metadata-eval 75.2%

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

   12. Simplified 75.2%

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

   if 1.4000000000000001e-7 < eps

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. +-rgt-identity 99.8%

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

      3. div-sub 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*r* 99.8%

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

      7. mul-1-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-neg-in 99.8%

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

      10. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 99.8%

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

       1. mul-1-neg 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in eps around inf 99.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 4: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{2}{e^{x}}}{2}\\ t_1 := x \cdot \varepsilon - x\\ t_2 := \frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{if}\;\varepsilon \leq 3 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{e^{t_1} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+267} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + \left(1 + t_1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 (exp x)) 2.0))
        (t_1 (- (* x eps) x))
        (t_2 (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0)))
   (if (<= eps 3e+17)
     t_0
     (if (<= eps 1.2e+65)
       t_2
       (if (<= eps 4.6e+120)
         t_0
         (if (<= eps 4.4e+138)
           (/ (+ (exp t_1) (- 1.0 (* x eps))) 2.0)
           (if (or (<= eps 3.2e+267) (not (<= eps 1.05e+299)))
             (/ (+ (exp (* x (- eps))) (+ 1.0 t_1)) 2.0)
             t_2)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (2.0 / exp(x)) / 2.0;
	double t_1 = (x * eps) - x;
	double t_2 = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	double tmp;
	if (eps <= 3e+17) {
		tmp = t_0;
	} else if (eps <= 1.2e+65) {
		tmp = t_2;
	} else if (eps <= 4.6e+120) {
		tmp = t_0;
	} else if (eps <= 4.4e+138) {
		tmp = (exp(t_1) + (1.0 - (x * eps))) / 2.0;
	} else if ((eps <= 3.2e+267) || !(eps <= 1.05e+299)) {
		tmp = (exp((x * -eps)) + (1.0 + t_1)) / 2.0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (2.0d0 / exp(x)) / 2.0d0
    t_1 = (x * eps) - x
    t_2 = (exp((x * (eps + (-1.0d0)))) - (-1.0d0)) / 2.0d0
    if (eps <= 3d+17) then
        tmp = t_0
    else if (eps <= 1.2d+65) then
        tmp = t_2
    else if (eps <= 4.6d+120) then
        tmp = t_0
    else if (eps <= 4.4d+138) then
        tmp = (exp(t_1) + (1.0d0 - (x * eps))) / 2.0d0
    else if ((eps <= 3.2d+267) .or. (.not. (eps <= 1.05d+299))) then
        tmp = (exp((x * -eps)) + (1.0d0 + t_1)) / 2.0d0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (2.0 / Math.exp(x)) / 2.0;
	double t_1 = (x * eps) - x;
	double t_2 = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	double tmp;
	if (eps <= 3e+17) {
		tmp = t_0;
	} else if (eps <= 1.2e+65) {
		tmp = t_2;
	} else if (eps <= 4.6e+120) {
		tmp = t_0;
	} else if (eps <= 4.4e+138) {
		tmp = (Math.exp(t_1) + (1.0 - (x * eps))) / 2.0;
	} else if ((eps <= 3.2e+267) || !(eps <= 1.05e+299)) {
		tmp = (Math.exp((x * -eps)) + (1.0 + t_1)) / 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (2.0 / math.exp(x)) / 2.0
	t_1 = (x * eps) - x
	t_2 = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	tmp = 0
	if eps <= 3e+17:
		tmp = t_0
	elif eps <= 1.2e+65:
		tmp = t_2
	elif eps <= 4.6e+120:
		tmp = t_0
	elif eps <= 4.4e+138:
		tmp = (math.exp(t_1) + (1.0 - (x * eps))) / 2.0
	elif (eps <= 3.2e+267) or not (eps <= 1.05e+299):
		tmp = (math.exp((x * -eps)) + (1.0 + t_1)) / 2.0
	else:
		tmp = t_2
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(2.0 / exp(x)) / 2.0)
	t_1 = Float64(Float64(x * eps) - x)
	t_2 = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0)
	tmp = 0.0
	if (eps <= 3e+17)
		tmp = t_0;
	elseif (eps <= 1.2e+65)
		tmp = t_2;
	elseif (eps <= 4.6e+120)
		tmp = t_0;
	elseif (eps <= 4.4e+138)
		tmp = Float64(Float64(exp(t_1) + Float64(1.0 - Float64(x * eps))) / 2.0);
	elseif ((eps <= 3.2e+267) || !(eps <= 1.05e+299))
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + Float64(1.0 + t_1)) / 2.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (2.0 / exp(x)) / 2.0;
	t_1 = (x * eps) - x;
	t_2 = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	tmp = 0.0;
	if (eps <= 3e+17)
		tmp = t_0;
	elseif (eps <= 1.2e+65)
		tmp = t_2;
	elseif (eps <= 4.6e+120)
		tmp = t_0;
	elseif (eps <= 4.4e+138)
		tmp = (exp(t_1) + (1.0 - (x * eps))) / 2.0;
	elseif ((eps <= 3.2e+267) || ~((eps <= 1.05e+299)))
		tmp = (exp((x * -eps)) + (1.0 + t_1)) / 2.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps, 3e+17], t$95$0, If[LessEqual[eps, 1.2e+65], t$95$2, If[LessEqual[eps, 4.6e+120], t$95$0, If[LessEqual[eps, 4.4e+138], N[(N[(N[Exp[t$95$1], $MachinePrecision] + N[(1.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps, 3.2e+267], N[Not[LessEqual[eps, 1.05e+299]], $MachinePrecision]], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 3e17 or 1.2000000000000001e65 < eps < 4.59999999999999985e120

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. mul-1-neg 98.3%

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

      4. associate-*r* 98.3%

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

      5. +-commutative 98.3%

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

      6. associate-*r* 98.3%

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

      7. mul-1-neg 98.3%

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

      8. *-commutative 98.3%

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

      9. distribute-rgt-neg-in 98.3%

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

      10. +-commutative 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in eps around 0 80.5%

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

      1. neg-mul-1 80.5%

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

   9. Simplified 80.5%

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

   10. Taylor expanded in eps around 0 76.5%

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

       1. exp-neg 76.5%

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

       2. associate-*r/ 76.5%

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

       3. metadata-eval 76.5%

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

   12. Simplified 76.5%

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

   if 3e17 < eps < 1.2000000000000001e65 or 3.2000000000000001e267 < eps < 1.05e299

   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. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 86.2%

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

   if 4.59999999999999985e120 < eps < 4.4000000000000001e138

   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. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf 100.0%

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

       1. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in eps around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. neg-mul-1 100.0%

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

       3. *-commutative 100.0%

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

       4. unsub-neg 100.0%

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

       5. *-commutative 100.0%

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

   15. Simplified 100.0%

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

   if 4.4000000000000001e138 < eps < 3.2000000000000001e267 or 1.05e299 < eps

   1. Initial program 99.7%

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

      1. div-sub 99.7%

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

      2. +-rgt-identity 99.7%

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

      3. div-sub 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in eps around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. mul-1-neg 99.7%

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

      4. associate-*r* 99.7%

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

      5. +-commutative 99.7%

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

      6. associate-*r* 99.7%

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

      7. mul-1-neg 99.7%

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

      8. *-commutative 99.7%

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

      9. distribute-rgt-neg-in 99.7%

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

      10. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in eps around inf 99.7%

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

      1. associate-*r* 99.7%

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

      2. neg-mul-1 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in x around inf 99.7%

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

       1. mul-1-neg 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in x around 0 72.1%

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

       1. *-commutative 72.1%

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

       2. distribute-lft-out-- 72.1%

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

       3. *-commutative 72.1%

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

       4. *-rgt-identity 72.1%

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

   15. Simplified 72.1%

       \[\leadsto \frac{\color{blue}{\left(1 + \left(\varepsilon \cdot x - x\right)\right)} + e^{-\varepsilon \cdot x}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{+138}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.2 \cdot 10^{+267} \lor \neg \left(\varepsilon \leq 1.05 \cdot 10^{+299}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + \left(1 + \left(x \cdot \varepsilon - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \end{array} \]

Alternative 5: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{2}{e^{x}}}{2}\\ t_1 := \frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{if}\;\varepsilon \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 (exp x)) 2.0))
        (t_1 (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0)))
   (if (<= eps 2.8e+17)
     t_0
     (if (<= eps 3.5e+64)
       t_1
       (if (<= eps 5.2e+120)
         t_0
         (if (<= eps 1.05e+138)
           (/ (+ (exp (- (* x eps) x)) (- 1.0 (* x eps))) 2.0)
           t_1))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (2.0 / exp(x)) / 2.0;
	double t_1 = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	double tmp;
	if (eps <= 2.8e+17) {
		tmp = t_0;
	} else if (eps <= 3.5e+64) {
		tmp = t_1;
	} else if (eps <= 5.2e+120) {
		tmp = t_0;
	} else if (eps <= 1.05e+138) {
		tmp = (exp(((x * eps) - x)) + (1.0 - (x * eps))) / 2.0;
	} 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 = (2.0d0 / exp(x)) / 2.0d0
    t_1 = (exp((x * (eps + (-1.0d0)))) - (-1.0d0)) / 2.0d0
    if (eps <= 2.8d+17) then
        tmp = t_0
    else if (eps <= 3.5d+64) then
        tmp = t_1
    else if (eps <= 5.2d+120) then
        tmp = t_0
    else if (eps <= 1.05d+138) then
        tmp = (exp(((x * eps) - x)) + (1.0d0 - (x * eps))) / 2.0d0
    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 = (2.0 / Math.exp(x)) / 2.0;
	double t_1 = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	double tmp;
	if (eps <= 2.8e+17) {
		tmp = t_0;
	} else if (eps <= 3.5e+64) {
		tmp = t_1;
	} else if (eps <= 5.2e+120) {
		tmp = t_0;
	} else if (eps <= 1.05e+138) {
		tmp = (Math.exp(((x * eps) - x)) + (1.0 - (x * eps))) / 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (2.0 / math.exp(x)) / 2.0
	t_1 = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	tmp = 0
	if eps <= 2.8e+17:
		tmp = t_0
	elif eps <= 3.5e+64:
		tmp = t_1
	elif eps <= 5.2e+120:
		tmp = t_0
	elif eps <= 1.05e+138:
		tmp = (math.exp(((x * eps) - x)) + (1.0 - (x * eps))) / 2.0
	else:
		tmp = t_1
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(2.0 / exp(x)) / 2.0)
	t_1 = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0)
	tmp = 0.0
	if (eps <= 2.8e+17)
		tmp = t_0;
	elseif (eps <= 3.5e+64)
		tmp = t_1;
	elseif (eps <= 5.2e+120)
		tmp = t_0;
	elseif (eps <= 1.05e+138)
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) + Float64(1.0 - Float64(x * eps))) / 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (2.0 / exp(x)) / 2.0;
	t_1 = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	tmp = 0.0;
	if (eps <= 2.8e+17)
		tmp = t_0;
	elseif (eps <= 3.5e+64)
		tmp = t_1;
	elseif (eps <= 5.2e+120)
		tmp = t_0;
	elseif (eps <= 1.05e+138)
		tmp = (exp(((x * eps) - x)) + (1.0 - (x * eps))) / 2.0;
	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[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps, 2.8e+17], t$95$0, If[LessEqual[eps, 3.5e+64], t$95$1, If[LessEqual[eps, 5.2e+120], t$95$0, If[LessEqual[eps, 1.05e+138], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 2.8e17 or 3.4999999999999999e64 < eps < 5.1999999999999998e120

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. mul-1-neg 98.3%

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

      4. associate-*r* 98.3%

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

      5. +-commutative 98.3%

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

      6. associate-*r* 98.3%

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

      7. mul-1-neg 98.3%

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

      8. *-commutative 98.3%

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

      9. distribute-rgt-neg-in 98.3%

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

      10. +-commutative 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in eps around 0 80.5%

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

      1. neg-mul-1 80.5%

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

   9. Simplified 80.5%

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

   10. Taylor expanded in eps around 0 76.5%

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

       1. exp-neg 76.5%

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

       2. associate-*r/ 76.5%

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

       3. metadata-eval 76.5%

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

   12. Simplified 76.5%

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

   if 2.8e17 < eps < 3.4999999999999999e64 or 1.05000000000000003e138 < eps

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. +-rgt-identity 99.8%

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

      3. div-sub 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*r* 99.8%

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

      7. mul-1-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-neg-in 99.8%

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

      10. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 71.2%

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

   if 5.1999999999999998e120 < eps < 1.05000000000000003e138

   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. Step-by-step derivation

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf 100.0%

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

       1. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in eps around 0 100.0%

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

       1. +-commutative 100.0%

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

       2. neg-mul-1 100.0%

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

       3. *-commutative 100.0%

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

       4. unsub-neg 100.0%

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

       5. *-commutative 100.0%

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

   15. Simplified 100.0%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{+138}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \end{array} \]

Alternative 6: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.8 \cdot 10^{+17} \lor \neg \left(\varepsilon \leq 1.2 \cdot 10^{+65}\right) \land \varepsilon \leq 1.06 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps 2.8e+17) (and (not (<= eps 1.2e+65)) (<= eps 1.06e+120)))
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((eps <= 2.8e+17) || (!(eps <= 1.2e+65) && (eps <= 1.06e+120))) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) - -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 <= 2.8d+17) .or. (.not. (eps <= 1.2d+65)) .and. (eps <= 1.06d+120)) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) - (-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 <= 2.8e+17) || (!(eps <= 1.2e+65) && (eps <= 1.06e+120))) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (eps <= 2.8e+17) or (not (eps <= 1.2e+65) and (eps <= 1.06e+120)):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((eps <= 2.8e+17) || (!(eps <= 1.2e+65) && (eps <= 1.06e+120)))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= 2.8e+17) || (~((eps <= 1.2e+65)) && (eps <= 1.06e+120)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[eps, 2.8e+17], And[N[Not[LessEqual[eps, 1.2e+65]], $MachinePrecision], LessEqual[eps, 1.06e+120]]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.8e17 or 1.2000000000000001e65 < eps < 1.05999999999999994e120

   1. Initial program 68.2%

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

      1. div-sub 68.2%

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

      2. +-rgt-identity 68.2%

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

      3. div-sub 68.2%

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

   3. Simplified 68.2%

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

   4. Taylor expanded in eps around inf 98.3%

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

      1. mul-1-neg 98.3%

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

      2. *-commutative 98.3%

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

      3. mul-1-neg 98.3%

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

      4. associate-*r* 98.3%

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

      5. +-commutative 98.3%

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

      6. associate-*r* 98.3%

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

      7. mul-1-neg 98.3%

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

      8. *-commutative 98.3%

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

      9. distribute-rgt-neg-in 98.3%

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

      10. +-commutative 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in eps around 0 80.5%

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

      1. neg-mul-1 80.5%

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

   9. Simplified 80.5%

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

   10. Taylor expanded in eps around 0 76.5%

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

       1. exp-neg 76.5%

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

       2. associate-*r/ 76.5%

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

       3. metadata-eval 76.5%

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

   12. Simplified 76.5%

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

   if 2.8e17 < eps < 1.2000000000000001e65 or 1.05999999999999994e120 < eps

   1. Initial program 99.8%

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

      1. div-sub 99.8%

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

      2. +-rgt-identity 99.8%

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

      3. div-sub 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-*r* 99.8%

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

      7. mul-1-neg 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-neg-in 99.8%

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

      10. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around 0 69.4%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.8 \cdot 10^{+17} \lor \neg \left(\varepsilon \leq 1.2 \cdot 10^{+65}\right) \land \varepsilon \leq 1.06 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \end{array} \]

Alternative 7: 71.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+166} \lor \neg \left(x \leq 5 \cdot 10^{+236}\right) \land x \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= x 5e+166) (and (not (<= x 5e+236)) (<= x 2e+251)))
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (* 0.3333333333333333 (pow x 3.0)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((x <= 5e+166) || (!(x <= 5e+236) && (x <= 2e+251))) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (0.3333333333333333 * pow(x, 3.0)) / 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 ((x <= 5d+166) .or. (.not. (x <= 5d+236)) .and. (x <= 2d+251)) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (0.3333333333333333d0 * (x ** 3.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if ((x <= 5e+166) || (!(x <= 5e+236) && (x <= 2e+251))) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (0.3333333333333333 * Math.pow(x, 3.0)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (x <= 5e+166) or (not (x <= 5e+236) and (x <= 2e+251)):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (0.3333333333333333 * math.pow(x, 3.0)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((x <= 5e+166) || (!(x <= 5e+236) && (x <= 2e+251)))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(0.3333333333333333 * (x ^ 3.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 5e+166) || (~((x <= 5e+236)) && (x <= 2e+251)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (0.3333333333333333 * (x ^ 3.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[x, 5e+166], And[N[Not[LessEqual[x, 5e+236]], $MachinePrecision], LessEqual[x, 2e+251]]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e166 or 4.9999999999999997e236 < x < 2.0000000000000001e251

   1. Initial program 72.0%

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

      1. div-sub 72.0%

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

      2. +-rgt-identity 72.0%

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

      3. div-sub 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in eps around inf 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. associate-*r* 98.5%

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

      5. +-commutative 98.5%

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

      6. associate-*r* 98.5%

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

      7. mul-1-neg 98.5%

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

      8. *-commutative 98.5%

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

      9. distribute-rgt-neg-in 98.5%

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

      10. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in eps around 0 85.7%

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

      1. neg-mul-1 85.7%

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

   9. Simplified 85.7%

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

   10. Taylor expanded in eps around 0 74.1%

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

       1. exp-neg 74.1%

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

       2. associate-*r/ 74.1%

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

       3. metadata-eval 74.1%

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

   12. Simplified 74.1%

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

   if 5.0000000000000002e166 < x < 4.9999999999999997e236 or 2.0000000000000001e251 < 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 0 29.2%

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

      1. *-commutative 29.2%

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

      2. distribute-lft1-in 29.2%

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

      3. mul-1-neg 29.2%

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

      4. distribute-lft-out 29.2%

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

      5. mul-1-neg 29.2%

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

      6. *-commutative 29.2%

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

      7. distribute-lft1-in 29.2%

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

      8. mul-1-neg 29.2%

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

   6. Simplified 29.2%

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

   7. Taylor expanded in x around 0 0.4%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
   8. Step-by-step derivation

      1. *-commutative 0.4%

         \[\leadsto \frac{\left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

      2. unpow2 0.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   9. Simplified 0.4%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   10. Taylor expanded in x around 0 0.0%

       \[\leadsto \frac{\color{blue}{2 + \left(-1 \cdot {x}^{2} + 0.3333333333333333 \cdot {x}^{3}\right)}}{2} \]
   11. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto \frac{2 + \color{blue}{\left(0.3333333333333333 \cdot {x}^{3} + -1 \cdot {x}^{2}\right)}}{2} \]

       2. associate-+r+ 0.0%

          \[\leadsto \frac{\color{blue}{\left(2 + 0.3333333333333333 \cdot {x}^{3}\right) + -1 \cdot {x}^{2}}}{2} \]

       3. mul-1-neg 0.0%

          \[\leadsto \frac{\left(2 + 0.3333333333333333 \cdot {x}^{3}\right) + \color{blue}{\left(-{x}^{2}\right)}}{2} \]

       4. unsub-neg 0.0%

          \[\leadsto \frac{\color{blue}{\left(2 + 0.3333333333333333 \cdot {x}^{3}\right) - {x}^{2}}}{2} \]

       5. *-commutative 0.0%

          \[\leadsto \frac{\left(2 + \color{blue}{{x}^{3} \cdot 0.3333333333333333}\right) - {x}^{2}}{2} \]

       6. unpow2 0.0%

          \[\leadsto \frac{\left(2 + {x}^{3} \cdot 0.3333333333333333\right) - \color{blue}{x \cdot x}}{2} \]

   12. Simplified 0.0%

       \[\leadsto \frac{\color{blue}{\left(2 + {x}^{3} \cdot 0.3333333333333333\right) - x \cdot x}}{2} \]

   13. Taylor expanded in x around inf 72.3%

       \[\leadsto \frac{\color{blue}{0.3333333333333333 \cdot {x}^{3}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+166} \lor \neg \left(x \leq 5 \cdot 10^{+236}\right) \land x \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot {x}^{3}}{2}\\ \end{array} \]

Alternative 8: 68.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 3.8e+138) (/ (/ 2.0 (exp x)) 2.0) (/ (+ 2.0 (* x eps)) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 3.8e+138) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * 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 <= 3.8d+138) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * 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 <= 3.8e+138) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 3.8e+138:
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (2.0 + (x * eps)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 3.8e+138)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 3.8e+138)
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (2.0 + (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 3.8e+138], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.80000000000000012e138

   1. Initial program 70.0%

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

      1. div-sub 70.0%

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

      2. +-rgt-identity 70.0%

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

      3. div-sub 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in eps around inf 98.4%

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

      1. mul-1-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. mul-1-neg 98.4%

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

      4. associate-*r* 98.4%

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

      5. +-commutative 98.4%

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

      6. associate-*r* 98.4%

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

      7. mul-1-neg 98.4%

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

      8. *-commutative 98.4%

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

      9. distribute-rgt-neg-in 98.4%

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

      10. +-commutative 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in eps around 0 81.6%

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

      1. neg-mul-1 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in eps around 0 75.0%

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

       1. exp-neg 75.0%

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

       2. associate-*r/ 75.0%

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

       3. metadata-eval 75.0%

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

   12. Simplified 75.0%

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

   if 3.80000000000000012e138 < eps

   1. Initial program 99.7%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x around 0 27.5%

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

   5. Taylor expanded in eps around 0 44.9%

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

   6. Taylor expanded in eps around 0 44.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 9: 63.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+151}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+238} \lor \neg \left(x \leq 1.85 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -180.0)
   (/ (* x (- -1.0 eps)) 2.0)
   (if (<= x 1.45)
     (/ (- 2.0 (* x x)) 2.0)
     (if (<= x 9.8e+151)
       0.0
       (if (or (<= x 3.2e+238) (not (<= x 1.85e+251)))
         (/
          (+ 2.0 (* x (+ (/ 1.0 eps) (* (- 1.0 eps) (+ -1.0 (/ -1.0 eps))))))
          2.0)
         0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -180.0) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 1.45) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 9.8e+151) {
		tmp = 0.0;
	} else if ((x <= 3.2e+238) || !(x <= 1.85e+251)) {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 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 <= (-180.0d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else if (x <= 1.45d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 9.8d+151) then
        tmp = 0.0d0
    else if ((x <= 3.2d+238) .or. (.not. (x <= 1.85d+251))) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps) + ((1.0d0 - eps) * ((-1.0d0) + ((-1.0d0) / eps)))))) / 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 <= -180.0) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 1.45) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 9.8e+151) {
		tmp = 0.0;
	} else if ((x <= 3.2e+238) || !(x <= 1.85e+251)) {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -180.0:
		tmp = (x * (-1.0 - eps)) / 2.0
	elif x <= 1.45:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 9.8e+151:
		tmp = 0.0
	elif (x <= 3.2e+238) or not (x <= 1.85e+251):
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -180.0)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	elseif (x <= 1.45)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 9.8e+151)
		tmp = 0.0;
	elseif ((x <= 3.2e+238) || !(x <= 1.85e+251))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(1.0 - eps) * Float64(-1.0 + Float64(-1.0 / eps)))))) / 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 <= -180.0)
		tmp = (x * (-1.0 - eps)) / 2.0;
	elseif (x <= 1.45)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 9.8e+151)
		tmp = 0.0;
	elseif ((x <= 3.2e+238) || ~((x <= 1.85e+251)))
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 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, -180.0], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.8e+151], 0.0, If[Or[LessEqual[x, 3.2e+238], N[Not[LessEqual[x, 1.85e+251]], $MachinePrecision]], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - eps), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -180

   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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. +-commutative 49.5%

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

      3. *-commutative 49.5%

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

      4. distribute-rgt-neg-in 49.5%

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

      5. +-commutative 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in x around inf 23.8%

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

   8. Taylor expanded in eps around inf 23.8%

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

      1. neg-mul-1 23.8%

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

   10. Simplified 23.8%

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

   if -180 < x < 1.44999999999999996

   1. Initial program 55.3%

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

      1. div-sub 55.3%

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

      2. +-rgt-identity 55.3%

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

      3. div-sub 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in eps around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. distribute-lft1-in 74.7%

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

      3. mul-1-neg 74.7%

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

      4. distribute-lft-out 74.7%

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

      5. mul-1-neg 74.7%

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

      6. *-commutative 74.7%

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

      7. distribute-lft1-in 74.7%

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

      8. mul-1-neg 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around 0 74.3%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
   8. Step-by-step derivation

      1. *-commutative 74.3%

         \[\leadsto \frac{\left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

      2. unpow2 74.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   9. Simplified 74.3%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   10. Taylor expanded in x around 0 74.3%

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

       1. mul-1-neg 74.3%

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

       2. unsub-neg 74.3%

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

       3. unpow2 74.3%

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

   12. Simplified 74.3%

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

   if 1.44999999999999996 < x < 9.7999999999999998e151 or 3.19999999999999981e238 < x < 1.84999999999999995e251

   1. Initial program 98.7%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in eps around 0 62.2%

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

      1. div-sub 62.2%

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

      2. rec-exp 62.2%

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

      3. mul-1-neg 62.2%

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

      4. +-inverses 62.2%

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

   6. Simplified 62.2%

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

   if 9.7999999999999998e151 < x < 3.19999999999999981e238 or 1.84999999999999995e251 < 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 x around 0 3.2%

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

   5. Taylor expanded in eps around 0 18.8%

      \[\leadsto \frac{\left(\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{\frac{-1}{\varepsilon}}\right) \cdot x + 2}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{+151}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+238} \lor \neg \left(x \leq 1.85 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 62.9% accurate, 13.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+156}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+235} \lor \neg \left(x \leq 2.2 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0)
   (/ (* x (- eps)) 2.0)
   (if (<= x 480.0)
     1.0
     (if (<= x 2.5e+156)
       0.0
       (if (or (<= x 6.8e+235) (not (<= x 2.2e+251)))
         (/ (+ 2.0 (* x eps)) 2.0)
         0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2.5e+156) {
		tmp = 0.0;
	} else if ((x <= 6.8e+235) || !(x <= 2.2e+251)) {
		tmp = (2.0 + (x * eps)) / 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 <= (-1.0d0)) then
        tmp = (x * -eps) / 2.0d0
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else if (x <= 2.5d+156) then
        tmp = 0.0d0
    else if ((x <= 6.8d+235) .or. (.not. (x <= 2.2d+251))) then
        tmp = (2.0d0 + (x * eps)) / 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 <= -1.0) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2.5e+156) {
		tmp = 0.0;
	} else if ((x <= 6.8e+235) || !(x <= 2.2e+251)) {
		tmp = (2.0 + (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (x * -eps) / 2.0
	elif x <= 480.0:
		tmp = 1.0
	elif x <= 2.5e+156:
		tmp = 0.0
	elif (x <= 6.8e+235) or not (x <= 2.2e+251):
		tmp = (2.0 + (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2.5e+156)
		tmp = 0.0;
	elseif ((x <= 6.8e+235) || !(x <= 2.2e+251))
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 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 <= -1.0)
		tmp = (x * -eps) / 2.0;
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2.5e+156)
		tmp = 0.0;
	elseif ((x <= 6.8e+235) || ~((x <= 2.2e+251)))
		tmp = (2.0 + (x * eps)) / 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, -1.0], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 2.5e+156], 0.0, If[Or[LessEqual[x, 6.8e+235], N[Not[LessEqual[x, 2.2e+251]], $MachinePrecision]], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. +-commutative 49.5%

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

      3. *-commutative 49.5%

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

      4. distribute-rgt-neg-in 49.5%

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

      5. +-commutative 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in x around inf 23.8%

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

   8. Taylor expanded in eps around inf 23.8%

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

      1. neg-mul-1 23.8%

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

      2. distribute-lft-neg-in 23.8%

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

      3. *-commutative 23.8%

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

   10. Simplified 23.8%

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

   if -1 < x < 480

   1. Initial program 55.5%

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

      1. div-sub 55.5%

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

      2. +-rgt-identity 55.5%

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

      3. div-sub 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 73.1%

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

   if 480 < x < 2.49999999999999996e156 or 6.79999999999999991e235 < x < 2.2e251

   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 64.6%

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

      1. div-sub 64.6%

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

      2. rec-exp 64.6%

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

      3. mul-1-neg 64.6%

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

      4. +-inverses 64.6%

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

   6. Simplified 64.6%

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

   if 2.49999999999999996e156 < x < 6.79999999999999991e235 or 2.2e251 < 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 x around 0 3.2%

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

   5. Taylor expanded in eps around 0 18.8%

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

   6. Taylor expanded in eps around 0 18.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+156}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+235} \lor \neg \left(x \leq 2.2 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 62.9% accurate, 13.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+236} \lor \neg \left(x \leq 2.25 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -50.0)
   (/ (* x (- eps)) 2.0)
   (if (<= x 1.45)
     (/ (- 2.0 (* x x)) 2.0)
     (if (<= x 1.02e+149)
       0.0
       (if (or (<= x 3e+236) (not (<= x 2.25e+251)))
         (/ (+ 2.0 (* x eps)) 2.0)
         0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -50.0) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 1.45) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 1.02e+149) {
		tmp = 0.0;
	} else if ((x <= 3e+236) || !(x <= 2.25e+251)) {
		tmp = (2.0 + (x * eps)) / 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 <= (-50.0d0)) then
        tmp = (x * -eps) / 2.0d0
    else if (x <= 1.45d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 1.02d+149) then
        tmp = 0.0d0
    else if ((x <= 3d+236) .or. (.not. (x <= 2.25d+251))) then
        tmp = (2.0d0 + (x * eps)) / 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 <= -50.0) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 1.45) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 1.02e+149) {
		tmp = 0.0;
	} else if ((x <= 3e+236) || !(x <= 2.25e+251)) {
		tmp = (2.0 + (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -50.0:
		tmp = (x * -eps) / 2.0
	elif x <= 1.45:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 1.02e+149:
		tmp = 0.0
	elif (x <= 3e+236) or not (x <= 2.25e+251):
		tmp = (2.0 + (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -50.0)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (x <= 1.45)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 1.02e+149)
		tmp = 0.0;
	elseif ((x <= 3e+236) || !(x <= 2.25e+251))
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 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 <= -50.0)
		tmp = (x * -eps) / 2.0;
	elseif (x <= 1.45)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 1.02e+149)
		tmp = 0.0;
	elseif ((x <= 3e+236) || ~((x <= 2.25e+251)))
		tmp = (2.0 + (x * eps)) / 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, -50.0], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.02e+149], 0.0, If[Or[LessEqual[x, 3e+236], N[Not[LessEqual[x, 2.25e+251]], $MachinePrecision]], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -50

   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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. +-commutative 49.5%

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

      3. *-commutative 49.5%

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

      4. distribute-rgt-neg-in 49.5%

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

      5. +-commutative 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in x around inf 23.8%

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

   8. Taylor expanded in eps around inf 23.8%

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

      1. neg-mul-1 23.8%

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

      2. distribute-lft-neg-in 23.8%

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

      3. *-commutative 23.8%

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

   10. Simplified 23.8%

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

   if -50 < x < 1.44999999999999996

   1. Initial program 55.3%

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

      1. div-sub 55.3%

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

      2. +-rgt-identity 55.3%

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

      3. div-sub 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in eps around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. distribute-lft1-in 74.7%

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

      3. mul-1-neg 74.7%

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

      4. distribute-lft-out 74.7%

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

      5. mul-1-neg 74.7%

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

      6. *-commutative 74.7%

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

      7. distribute-lft1-in 74.7%

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

      8. mul-1-neg 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around 0 74.3%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
   8. Step-by-step derivation

      1. *-commutative 74.3%

         \[\leadsto \frac{\left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

      2. unpow2 74.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   9. Simplified 74.3%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   10. Taylor expanded in x around 0 74.3%

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

       1. mul-1-neg 74.3%

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

       2. unsub-neg 74.3%

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

       3. unpow2 74.3%

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

   12. Simplified 74.3%

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

   if 1.44999999999999996 < x < 1.01999999999999997e149 or 2.9999999999999998e236 < x < 2.2499999999999999e251

   1. Initial program 98.7%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in eps around 0 62.2%

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

      1. div-sub 62.2%

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

      2. rec-exp 62.2%

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

      3. mul-1-neg 62.2%

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

      4. +-inverses 62.2%

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

   6. Simplified 62.2%

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

   if 1.01999999999999997e149 < x < 2.9999999999999998e236 or 2.2499999999999999e251 < 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 x around 0 3.2%

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

   5. Taylor expanded in eps around 0 18.8%

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

   6. Taylor expanded in eps around 0 18.6%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+149}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+236} \lor \neg \left(x \leq 2.25 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 63.0% accurate, 13.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -300:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+151}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+238} \lor \neg \left(x \leq 2.5 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -300.0)
   (/ (* x (- -1.0 eps)) 2.0)
   (if (<= x 1.45)
     (/ (- 2.0 (* x x)) 2.0)
     (if (<= x 2.45e+151)
       0.0
       (if (or (<= x 2.55e+238) (not (<= x 2.5e+251)))
         (/ (+ 2.0 (* x eps)) 2.0)
         0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -300.0) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 1.45) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 2.45e+151) {
		tmp = 0.0;
	} else if ((x <= 2.55e+238) || !(x <= 2.5e+251)) {
		tmp = (2.0 + (x * eps)) / 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 <= (-300.0d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else if (x <= 1.45d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 2.45d+151) then
        tmp = 0.0d0
    else if ((x <= 2.55d+238) .or. (.not. (x <= 2.5d+251))) then
        tmp = (2.0d0 + (x * eps)) / 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 <= -300.0) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 1.45) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 2.45e+151) {
		tmp = 0.0;
	} else if ((x <= 2.55e+238) || !(x <= 2.5e+251)) {
		tmp = (2.0 + (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -300.0:
		tmp = (x * (-1.0 - eps)) / 2.0
	elif x <= 1.45:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 2.45e+151:
		tmp = 0.0
	elif (x <= 2.55e+238) or not (x <= 2.5e+251):
		tmp = (2.0 + (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -300.0)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	elseif (x <= 1.45)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 2.45e+151)
		tmp = 0.0;
	elseif ((x <= 2.55e+238) || !(x <= 2.5e+251))
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 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 <= -300.0)
		tmp = (x * (-1.0 - eps)) / 2.0;
	elseif (x <= 1.45)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 2.45e+151)
		tmp = 0.0;
	elseif ((x <= 2.55e+238) || ~((x <= 2.5e+251)))
		tmp = (2.0 + (x * eps)) / 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, -300.0], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.45e+151], 0.0, If[Or[LessEqual[x, 2.55e+238], N[Not[LessEqual[x, 2.5e+251]], $MachinePrecision]], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -300

   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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. +-commutative 49.5%

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

      3. *-commutative 49.5%

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

      4. distribute-rgt-neg-in 49.5%

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

      5. +-commutative 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in x around inf 23.8%

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

   8. Taylor expanded in eps around inf 23.8%

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

      1. neg-mul-1 23.8%

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

   10. Simplified 23.8%

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

   if -300 < x < 1.44999999999999996

   1. Initial program 55.3%

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

      1. div-sub 55.3%

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

      2. +-rgt-identity 55.3%

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

      3. div-sub 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in eps around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. distribute-lft1-in 74.7%

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

      3. mul-1-neg 74.7%

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

      4. distribute-lft-out 74.7%

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

      5. mul-1-neg 74.7%

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

      6. *-commutative 74.7%

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

      7. distribute-lft1-in 74.7%

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

      8. mul-1-neg 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in x around 0 74.3%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
   8. Step-by-step derivation

      1. *-commutative 74.3%

         \[\leadsto \frac{\left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

      2. unpow2 74.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   9. Simplified 74.3%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.5\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   10. Taylor expanded in x around 0 74.3%

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

       1. mul-1-neg 74.3%

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

       2. unsub-neg 74.3%

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

       3. unpow2 74.3%

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

   12. Simplified 74.3%

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

   if 1.44999999999999996 < x < 2.45e151 or 2.5500000000000001e238 < x < 2.5000000000000002e251

   1. Initial program 98.7%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in eps around 0 62.2%

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

      1. div-sub 62.2%

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

      2. rec-exp 62.2%

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

      3. mul-1-neg 62.2%

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

      4. +-inverses 62.2%

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

   6. Simplified 62.2%

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

   if 2.45e151 < x < 2.5500000000000001e238 or 2.5000000000000002e251 < 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 x around 0 3.2%

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

   5. Taylor expanded in eps around 0 18.8%

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

   6. Taylor expanded in eps around 0 18.6%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -300:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.45 \cdot 10^{+151}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.55 \cdot 10^{+238} \lor \neg \left(x \leq 2.5 \cdot 10^{+251}\right):\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 64.1% accurate, 28.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0) (/ (* x (- eps)) 2.0) (if (<= x 520.0) 1.0 0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 520.0) {
		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 <= (-1.0d0)) then
        tmp = (x * -eps) / 2.0d0
    else if (x <= 520.0d0) 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 <= -1.0) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (x * -eps) / 2.0
	elif x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (x <= 520.0)
		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 <= -1.0)
		tmp = (x * -eps) / 2.0;
	elseif (x <= 520.0)
		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, -1.0], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   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 49.5%

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

      1. mul-1-neg 49.5%

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

      2. +-commutative 49.5%

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

      3. *-commutative 49.5%

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

      4. distribute-rgt-neg-in 49.5%

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

      5. +-commutative 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in x around inf 23.8%

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

   8. Taylor expanded in eps around inf 23.8%

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

      1. neg-mul-1 23.8%

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

      2. distribute-lft-neg-in 23.8%

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

      3. *-commutative 23.8%

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

   10. Simplified 23.8%

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

   if -1 < x < 520

   1. Initial program 55.5%

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

      1. div-sub 55.5%

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

      2. +-rgt-identity 55.5%

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

      3. div-sub 55.5%

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

   3. Simplified 55.5%

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

   4. Taylor expanded in x around 0 73.1%

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

   if 520 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 50.8%

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

      1. div-sub 50.8%

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

      2. rec-exp 50.8%

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

      3. mul-1-neg 50.8%

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

      4. +-inverses 50.8%

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

   6. Simplified 50.8%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 57.0% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		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 <= 550.0d0) 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 <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		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 <= 550.0)
		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, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 63.5%

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

      1. div-sub 63.5%

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

      2. +-rgt-identity 63.5%

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

      3. div-sub 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in x around 0 60.4%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 50.8%

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

      1. div-sub 50.8%

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

      2. rec-exp 50.8%

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

      3. mul-1-neg 50.8%

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

      4. +-inverses 50.8%

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

   6. Simplified 50.8%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 15.9% 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 75.5%

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

3. Simplified 68.5%

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

4. Taylor expanded in eps around 0 18.2%

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

   1. div-sub 18.2%

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

   2. rec-exp 18.2%

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

   3. mul-1-neg 18.2%

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

   4. +-inverses 18.3%

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

6. Simplified 18.3%

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

7. Final simplification 18.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023230 
(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))