NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 98.8%

Time: 13.1s

Alternatives: 15

Speedup: 2.0×

• 38.4% 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: 73.3% 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: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 3.8)
   (/ (+ (exp (- (* x eps) x)) (exp (* x (- eps)))) 2.0)
   (/ (+ (exp (* x (+ eps -1.0))) (exp (- x))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 3.8) {
		tmp = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + 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 <= 3.8d0) then
        tmp = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + 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 <= 3.8) {
		tmp = (Math.exp(((x * eps) - x)) + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 3.8:
		tmp = (math.exp(((x * eps) - x)) + math.exp((x * -eps))) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 3.8)
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.8)
		tmp = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + 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, 3.8], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7999999999999998

   1. Initial program 62.9%

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

      1. div-sub 62.9%

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

      2. +-rgt-identity 62.9%

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

      3. div-sub 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in eps around inf 98.6%

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

      1. *-commutative 98.6%

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

      2. sub-neg 98.6%

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

      3. mul-1-neg 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. *-commutative 98.6%

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

      7. mul-1-neg 98.6%

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

      8. sub-neg 98.6%

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

      9. mul-1-neg 98.6%

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

      10. +-commutative 98.6%

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

      11. exp-prod 98.6%

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

      12. *-lft-identity 98.6%

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

      13. metadata-eval 98.6%

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

      14. cancel-sign-sub-inv 98.6%

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

      15. exp-prod 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in eps around inf 98.6%

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

   8. Taylor expanded in x around inf 98.6%

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

   if 3.7999999999999998 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

      12. *-lft-identity 100.0%

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

      13. metadata-eval 100.0%

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

      14. cancel-sign-sub-inv 100.0%

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

      15. exp-prod 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 65.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \]

Alternative 2: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + \left(1 - x \cdot \left(1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2.45e+86)
   (/ (/ 2.0 (exp x)) 2.0)
   (if (<= x -4.8e-167)
     (/ (+ (exp (* x (- eps))) (- 1.0 (* x (- 1.0 eps)))) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (exp (- x))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2.45e+86) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else if (x <= -4.8e-167) {
		tmp = (exp((x * -eps)) + (1.0 - (x * (1.0 - eps)))) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + 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 <= (-2.45d+86)) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else if (x <= (-4.8d-167)) then
        tmp = (exp((x * -eps)) + (1.0d0 - (x * (1.0d0 - eps)))) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + 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 <= -2.45e+86) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else if (x <= -4.8e-167) {
		tmp = (Math.exp((x * -eps)) + (1.0 - (x * (1.0 - eps)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -2.45e+86:
		tmp = (2.0 / math.exp(x)) / 2.0
	elif x <= -4.8e-167:
		tmp = (math.exp((x * -eps)) + (1.0 - (x * (1.0 - eps)))) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -2.45e+86)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	elseif (x <= -4.8e-167)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + Float64(1.0 - Float64(x * Float64(1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.45e+86)
		tmp = (2.0 / exp(x)) / 2.0;
	elseif (x <= -4.8e-167)
		tmp = (exp((x * -eps)) + (1.0 - (x * (1.0 - eps)))) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + 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, -2.45e+86], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.8e-167], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.45e86

   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. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

      12. *-lft-identity 100.0%

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

      13. metadata-eval 100.0%

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

      14. cancel-sign-sub-inv 100.0%

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

      15. exp-prod 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. exp-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   if -2.45e86 < x < -4.79999999999999986e-167

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in eps around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. sub-neg 97.5%

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

      3. mul-1-neg 97.5%

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

      4. *-commutative 97.5%

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

      5. mul-1-neg 97.5%

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

      6. *-commutative 97.5%

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

      7. mul-1-neg 97.5%

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

      8. sub-neg 97.5%

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

      9. mul-1-neg 97.5%

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

      10. +-commutative 97.5%

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

      11. exp-prod 97.5%

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

      12. *-lft-identity 97.5%

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

      13. metadata-eval 97.5%

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

      14. cancel-sign-sub-inv 97.5%

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

      15. exp-prod 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in eps around inf 97.5%

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

   8. Taylor expanded in x around 0 77.8%

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

   if -4.79999999999999986e-167 < x

   1. Initial program 73.4%

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

      1. div-sub 73.4%

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

      2. +-rgt-identity 73.4%

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

      3. div-sub 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in eps around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. sub-neg 99.2%

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

      3. mul-1-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. mul-1-neg 99.2%

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

      6. *-commutative 99.2%

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

      7. mul-1-neg 99.2%

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

      8. sub-neg 99.2%

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

      9. mul-1-neg 99.2%

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

      10. +-commutative 99.2%

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

      11. exp-prod 99.2%

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

      12. *-lft-identity 99.2%

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

      13. metadata-eval 99.2%

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

      14. cancel-sign-sub-inv 99.2%

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

      15. exp-prod 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in eps around 0 82.2%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + \left(1 - x \cdot \left(1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \]

Alternative 3: 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 74.2%

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

   1. div-sub 74.2%

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

   2. +-rgt-identity 74.2%

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

   3. div-sub 74.2%

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

3. Simplified 74.2%

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

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. *-commutative 99.0%

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

   2. sub-neg 99.0%

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

   3. mul-1-neg 99.0%

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

   4. *-commutative 99.0%

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

   5. mul-1-neg 99.0%

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

   6. *-commutative 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. mul-1-neg 99.0%

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

   10. +-commutative 99.0%

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

   11. exp-prod 99.0%

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

   12. *-lft-identity 99.0%

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

   13. metadata-eval 99.0%

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

   14. cancel-sign-sub-inv 99.0%

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

   15. exp-prod 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 4: 85.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 - x \cdot \left(1 - \varepsilon\right)\\ t_1 := e^{x \cdot \left(\varepsilon + -1\right)}\\ t_2 := \frac{\frac{2}{e^{x}}}{2}\\ t_3 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + t_0}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{t_1 - -1}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+223}:\\ \;\;\;\;\frac{t_1 \cdot t_3 + \left(\left(1 - x\right) + \left(x - x \cdot \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+259}:\\ \;\;\;\;\frac{t_0 \cdot t_3 + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (- 1.0 eps))))
        (t_1 (exp (* x (+ eps -1.0))))
        (t_2 (/ (/ 2.0 (exp x)) 2.0))
        (t_3 (+ 1.0 (/ 1.0 eps))))
   (if (<= x -2.45e+86)
     t_2
     (if (<= x -4.8e-167)
       (/ (+ (exp (* x (- eps))) t_0) 2.0)
       (if (<= x 1.25e+61)
         (/ (- t_1 -1.0) 2.0)
         (if (<= x 1.85e+223)
           (/ (+ (* t_1 t_3) (+ (- 1.0 x) (- x (* x eps)))) 2.0)
           (if (<= x 3.5e+259)
             (/
              (+ (* t_0 t_3) (* (+ 1.0 (* x (- -1.0 eps))) (/ -1.0 eps)))
              2.0)
             t_2)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 - (x * (1.0 - eps));
	double t_1 = exp((x * (eps + -1.0)));
	double t_2 = (2.0 / exp(x)) / 2.0;
	double t_3 = 1.0 + (1.0 / eps);
	double tmp;
	if (x <= -2.45e+86) {
		tmp = t_2;
	} else if (x <= -4.8e-167) {
		tmp = (exp((x * -eps)) + t_0) / 2.0;
	} else if (x <= 1.25e+61) {
		tmp = (t_1 - -1.0) / 2.0;
	} else if (x <= 1.85e+223) {
		tmp = ((t_1 * t_3) + ((1.0 - x) + (x - (x * eps)))) / 2.0;
	} else if (x <= 3.5e+259) {
		tmp = ((t_0 * t_3) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 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) :: t_3
    real(8) :: tmp
    t_0 = 1.0d0 - (x * (1.0d0 - eps))
    t_1 = exp((x * (eps + (-1.0d0))))
    t_2 = (2.0d0 / exp(x)) / 2.0d0
    t_3 = 1.0d0 + (1.0d0 / eps)
    if (x <= (-2.45d+86)) then
        tmp = t_2
    else if (x <= (-4.8d-167)) then
        tmp = (exp((x * -eps)) + t_0) / 2.0d0
    else if (x <= 1.25d+61) then
        tmp = (t_1 - (-1.0d0)) / 2.0d0
    else if (x <= 1.85d+223) then
        tmp = ((t_1 * t_3) + ((1.0d0 - x) + (x - (x * eps)))) / 2.0d0
    else if (x <= 3.5d+259) then
        tmp = ((t_0 * t_3) + ((1.0d0 + (x * ((-1.0d0) - eps))) * ((-1.0d0) / eps))) / 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 = 1.0 - (x * (1.0 - eps));
	double t_1 = Math.exp((x * (eps + -1.0)));
	double t_2 = (2.0 / Math.exp(x)) / 2.0;
	double t_3 = 1.0 + (1.0 / eps);
	double tmp;
	if (x <= -2.45e+86) {
		tmp = t_2;
	} else if (x <= -4.8e-167) {
		tmp = (Math.exp((x * -eps)) + t_0) / 2.0;
	} else if (x <= 1.25e+61) {
		tmp = (t_1 - -1.0) / 2.0;
	} else if (x <= 1.85e+223) {
		tmp = ((t_1 * t_3) + ((1.0 - x) + (x - (x * eps)))) / 2.0;
	} else if (x <= 3.5e+259) {
		tmp = ((t_0 * t_3) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 - (x * (1.0 - eps))
	t_1 = math.exp((x * (eps + -1.0)))
	t_2 = (2.0 / math.exp(x)) / 2.0
	t_3 = 1.0 + (1.0 / eps)
	tmp = 0
	if x <= -2.45e+86:
		tmp = t_2
	elif x <= -4.8e-167:
		tmp = (math.exp((x * -eps)) + t_0) / 2.0
	elif x <= 1.25e+61:
		tmp = (t_1 - -1.0) / 2.0
	elif x <= 1.85e+223:
		tmp = ((t_1 * t_3) + ((1.0 - x) + (x - (x * eps)))) / 2.0
	elif x <= 3.5e+259:
		tmp = ((t_0 * t_3) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0
	else:
		tmp = t_2
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 - Float64(x * Float64(1.0 - eps)))
	t_1 = exp(Float64(x * Float64(eps + -1.0)))
	t_2 = Float64(Float64(2.0 / exp(x)) / 2.0)
	t_3 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (x <= -2.45e+86)
		tmp = t_2;
	elseif (x <= -4.8e-167)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + t_0) / 2.0);
	elseif (x <= 1.25e+61)
		tmp = Float64(Float64(t_1 - -1.0) / 2.0);
	elseif (x <= 1.85e+223)
		tmp = Float64(Float64(Float64(t_1 * t_3) + Float64(Float64(1.0 - x) + Float64(x - Float64(x * eps)))) / 2.0);
	elseif (x <= 3.5e+259)
		tmp = Float64(Float64(Float64(t_0 * t_3) + Float64(Float64(1.0 + Float64(x * Float64(-1.0 - eps))) * Float64(-1.0 / eps))) / 2.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 - (x * (1.0 - eps));
	t_1 = exp((x * (eps + -1.0)));
	t_2 = (2.0 / exp(x)) / 2.0;
	t_3 = 1.0 + (1.0 / eps);
	tmp = 0.0;
	if (x <= -2.45e+86)
		tmp = t_2;
	elseif (x <= -4.8e-167)
		tmp = (exp((x * -eps)) + t_0) / 2.0;
	elseif (x <= 1.25e+61)
		tmp = (t_1 - -1.0) / 2.0;
	elseif (x <= 1.85e+223)
		tmp = ((t_1 * t_3) + ((1.0 - x) + (x - (x * eps)))) / 2.0;
	elseif (x <= 3.5e+259)
		tmp = ((t_0 * t_3) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 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[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.45e+86], t$95$2, If[LessEqual[x, -4.8e-167], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.25e+61], N[(N[(t$95$1 - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.85e+223], N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] + N[(x - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.5e+259], N[(N[(N[(t$95$0 * t$95$3), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$2]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.45e86 or 3.4999999999999998e259 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

      12. *-lft-identity 100.0%

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

      13. metadata-eval 100.0%

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

      14. cancel-sign-sub-inv 100.0%

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

      15. exp-prod 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 89.5%

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

   8. Taylor expanded in eps around 0 89.5%

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

      1. exp-neg 89.5%

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

      2. associate-*r/ 89.5%

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

      3. metadata-eval 89.5%

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

   10. Simplified 89.5%

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

   if -2.45e86 < x < -4.79999999999999986e-167

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in eps around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. sub-neg 97.5%

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

      3. mul-1-neg 97.5%

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

      4. *-commutative 97.5%

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

      5. mul-1-neg 97.5%

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

      6. *-commutative 97.5%

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

      7. mul-1-neg 97.5%

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

      8. sub-neg 97.5%

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

      9. mul-1-neg 97.5%

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

      10. +-commutative 97.5%

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

      11. exp-prod 97.5%

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

      12. *-lft-identity 97.5%

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

      13. metadata-eval 97.5%

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

      14. cancel-sign-sub-inv 97.5%

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

      15. exp-prod 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in eps around inf 97.5%

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

   8. Taylor expanded in x around 0 77.8%

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

   if -4.79999999999999986e-167 < x < 1.25000000000000004e61

   1. Initial program 57.9%

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

      1. div-sub 57.9%

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

      2. +-rgt-identity 57.9%

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

      3. div-sub 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

      2. sub-neg 98.8%

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

      3. mul-1-neg 98.8%

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

      4. *-commutative 98.8%

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

      5. mul-1-neg 98.8%

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

      6. *-commutative 98.8%

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

      7. mul-1-neg 98.8%

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

      8. sub-neg 98.8%

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

      9. mul-1-neg 98.8%

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

      10. +-commutative 98.8%

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

      11. exp-prod 98.8%

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

      12. *-lft-identity 98.8%

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

      13. metadata-eval 98.8%

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

      14. cancel-sign-sub-inv 98.8%

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

      15. exp-prod 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around 0 88.2%

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

   if 1.25000000000000004e61 < x < 1.8500000000000001e223

   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 28.2%

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

      1. mul-1-neg 28.2%

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

      2. +-commutative 28.2%

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

      3. *-commutative 28.2%

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

      4. +-commutative 28.2%

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

   6. Simplified 28.2%

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

   7. Taylor expanded in eps around inf 72.7%

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

      1. +-commutative 72.7%

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

      2. distribute-rgt-out 72.7%

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

      3. metadata-eval 72.7%

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

      4. sub-neg 72.7%

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

      5. *-commutative 72.7%

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

      6. mul-1-neg 72.7%

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

      7. unsub-neg 72.7%

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

      8. *-commutative 72.7%

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

      9. distribute-rgt-out-- 72.7%

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

      10. *-lft-identity 72.7%

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

   9. Simplified 72.7%

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

   if 1.8500000000000001e223 < x < 3.4999999999999998e259

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 45.2%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right) + 1\right) - \color{blue}{\frac{1}{\varepsilon}} \cdot \left(1 + \left(-x \cdot \left(\varepsilon + 1\right)\right)\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + \left(1 - x \cdot \left(1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+61}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+223}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\left(1 - x\right) + \left(x - x \cdot \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+259}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \end{array} \]

Alternative 5: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 - x \cdot \left(1 - \varepsilon\right)\\ t_1 := \frac{\frac{2}{e^{x}}}{2}\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + t_0}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+158}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+222} \lor \neg \left(x \leq 2.8 \cdot 10^{+259}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (- 1.0 eps)))) (t_1 (/ (/ 2.0 (exp x)) 2.0)))
   (if (<= x -2.45e+86)
     t_1
     (if (<= x -4.8e-167)
       (/ (+ (exp (* x (- eps))) t_0) 2.0)
       (if (<= x 3.7e+158)
         (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0)
         (if (or (<= x 5.5e+222) (not (<= x 2.8e+259)))
           t_1
           (/
            (+
             (* t_0 (+ 1.0 (/ 1.0 eps)))
             (* (+ 1.0 (* x (- -1.0 eps))) (/ -1.0 eps)))
            2.0)))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 - (x * (1.0 - eps));
	double t_1 = (2.0 / exp(x)) / 2.0;
	double tmp;
	if (x <= -2.45e+86) {
		tmp = t_1;
	} else if (x <= -4.8e-167) {
		tmp = (exp((x * -eps)) + t_0) / 2.0;
	} else if (x <= 3.7e+158) {
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	} else if ((x <= 5.5e+222) || !(x <= 2.8e+259)) {
		tmp = t_1;
	} else {
		tmp = ((t_0 * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x * (1.0d0 - eps))
    t_1 = (2.0d0 / exp(x)) / 2.0d0
    if (x <= (-2.45d+86)) then
        tmp = t_1
    else if (x <= (-4.8d-167)) then
        tmp = (exp((x * -eps)) + t_0) / 2.0d0
    else if (x <= 3.7d+158) then
        tmp = (exp((x * (eps + (-1.0d0)))) - (-1.0d0)) / 2.0d0
    else if ((x <= 5.5d+222) .or. (.not. (x <= 2.8d+259))) then
        tmp = t_1
    else
        tmp = ((t_0 * (1.0d0 + (1.0d0 / eps))) + ((1.0d0 + (x * ((-1.0d0) - eps))) * ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 - (x * (1.0 - eps));
	double t_1 = (2.0 / Math.exp(x)) / 2.0;
	double tmp;
	if (x <= -2.45e+86) {
		tmp = t_1;
	} else if (x <= -4.8e-167) {
		tmp = (Math.exp((x * -eps)) + t_0) / 2.0;
	} else if (x <= 3.7e+158) {
		tmp = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	} else if ((x <= 5.5e+222) || !(x <= 2.8e+259)) {
		tmp = t_1;
	} else {
		tmp = ((t_0 * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 - (x * (1.0 - eps))
	t_1 = (2.0 / math.exp(x)) / 2.0
	tmp = 0
	if x <= -2.45e+86:
		tmp = t_1
	elif x <= -4.8e-167:
		tmp = (math.exp((x * -eps)) + t_0) / 2.0
	elif x <= 3.7e+158:
		tmp = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	elif (x <= 5.5e+222) or not (x <= 2.8e+259):
		tmp = t_1
	else:
		tmp = ((t_0 * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 - Float64(x * Float64(1.0 - eps)))
	t_1 = Float64(Float64(2.0 / exp(x)) / 2.0)
	tmp = 0.0
	if (x <= -2.45e+86)
		tmp = t_1;
	elseif (x <= -4.8e-167)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + t_0) / 2.0);
	elseif (x <= 3.7e+158)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0);
	elseif ((x <= 5.5e+222) || !(x <= 2.8e+259))
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(t_0 * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(1.0 + Float64(x * Float64(-1.0 - eps))) * Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 - (x * (1.0 - eps));
	t_1 = (2.0 / exp(x)) / 2.0;
	tmp = 0.0;
	if (x <= -2.45e+86)
		tmp = t_1;
	elseif (x <= -4.8e-167)
		tmp = (exp((x * -eps)) + t_0) / 2.0;
	elseif (x <= 3.7e+158)
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	elseif ((x <= 5.5e+222) || ~((x <= 2.8e+259)))
		tmp = t_1;
	else
		tmp = ((t_0 * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.45e+86], t$95$1, If[LessEqual[x, -4.8e-167], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.7e+158], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5.5e+222], N[Not[LessEqual[x, 2.8e+259]], $MachinePrecision]], t$95$1, N[(N[(N[(t$95$0 * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.45e86 or 3.70000000000000011e158 < x < 5.4999999999999999e222 or 2.8000000000000001e259 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. *-commutative 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. +-commutative 100.0%

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

      11. exp-prod 100.0%

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

      12. *-lft-identity 100.0%

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

      13. metadata-eval 100.0%

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

      14. cancel-sign-sub-inv 100.0%

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

      15. exp-prod 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 91.5%

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

   8. Taylor expanded in eps around 0 88.1%

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

      1. exp-neg 88.1%

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

      2. associate-*r/ 88.1%

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

      3. metadata-eval 88.1%

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

   10. Simplified 88.1%

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

   if -2.45e86 < x < -4.79999999999999986e-167

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in eps around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. sub-neg 97.5%

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

      3. mul-1-neg 97.5%

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

      4. *-commutative 97.5%

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

      5. mul-1-neg 97.5%

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

      6. *-commutative 97.5%

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

      7. mul-1-neg 97.5%

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

      8. sub-neg 97.5%

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

      9. mul-1-neg 97.5%

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

      10. +-commutative 97.5%

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

      11. exp-prod 97.5%

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

      12. *-lft-identity 97.5%

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

      13. metadata-eval 97.5%

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

      14. cancel-sign-sub-inv 97.5%

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

      15. exp-prod 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in eps around inf 97.5%

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

   8. Taylor expanded in x around 0 77.8%

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

   if -4.79999999999999986e-167 < x < 3.70000000000000011e158

   1. Initial program 66.8%

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

      1. div-sub 66.8%

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

      2. +-rgt-identity 66.8%

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

      3. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in eps around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. mul-1-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. mul-1-neg 99.1%

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

      6. *-commutative 99.1%

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

      7. mul-1-neg 99.1%

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

      8. sub-neg 99.1%

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

      9. mul-1-neg 99.1%

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

      10. +-commutative 99.1%

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

      11. exp-prod 99.1%

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

      12. *-lft-identity 99.1%

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

      13. metadata-eval 99.1%

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

      14. cancel-sign-sub-inv 99.1%

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

      15. exp-prod 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in x around 0 74.1%

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

   if 5.4999999999999999e222 < x < 2.8000000000000001e259

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 45.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + \left(1 - x \cdot \left(1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+158}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+222} \lor \neg \left(x \leq 2.8 \cdot 10^{+259}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \]

Alternative 6: 69.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{2}{e^{x}}}{2}\\ \mathbf{if}\;x \leq 5000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+222} \lor \neg \left(x \leq 1.05 \cdot 10^{+258}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 (exp x)) 2.0)))
   (if (<= x 5000000000.0)
     t_0
     (if (<= x 1.5e+158)
       (/ (/ (+ x (expm1 x)) eps) 2.0)
       (if (or (<= x 5.1e+222) (not (<= x 1.05e+258)))
         t_0
         (/
          (+
           (* (- 1.0 (* x (- 1.0 eps))) (+ 1.0 (/ 1.0 eps)))
           (* (+ 1.0 (* x (- -1.0 eps))) (/ -1.0 eps)))
          2.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (2.0 / exp(x)) / 2.0;
	double tmp;
	if (x <= 5000000000.0) {
		tmp = t_0;
	} else if (x <= 1.5e+158) {
		tmp = ((x + expm1(x)) / eps) / 2.0;
	} else if ((x <= 5.1e+222) || !(x <= 1.05e+258)) {
		tmp = t_0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (2.0 / Math.exp(x)) / 2.0;
	double tmp;
	if (x <= 5000000000.0) {
		tmp = t_0;
	} else if (x <= 1.5e+158) {
		tmp = ((x + Math.expm1(x)) / eps) / 2.0;
	} else if ((x <= 5.1e+222) || !(x <= 1.05e+258)) {
		tmp = t_0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (2.0 / math.exp(x)) / 2.0
	tmp = 0
	if x <= 5000000000.0:
		tmp = t_0
	elif x <= 1.5e+158:
		tmp = ((x + math.expm1(x)) / eps) / 2.0
	elif (x <= 5.1e+222) or not (x <= 1.05e+258):
		tmp = t_0
	else:
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(2.0 / exp(x)) / 2.0)
	tmp = 0.0
	if (x <= 5000000000.0)
		tmp = t_0;
	elseif (x <= 1.5e+158)
		tmp = Float64(Float64(Float64(x + expm1(x)) / eps) / 2.0);
	elseif ((x <= 5.1e+222) || !(x <= 1.05e+258))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps))) * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(1.0 + Float64(x * Float64(-1.0 - eps))) * Float64(-1.0 / eps))) / 2.0);
	end
	return 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]}, If[LessEqual[x, 5000000000.0], t$95$0, If[LessEqual[x, 1.5e+158], N[(N[(N[(x + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5.1e+222], N[Not[LessEqual[x, 1.05e+258]], $MachinePrecision]], t$95$0, N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5e9 or 1.5e158 < x < 5.0999999999999999e222 or 1.04999999999999998e258 < x

   1. Initial program 68.0%

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

      1. div-sub 68.0%

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

      2. +-rgt-identity 68.0%

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

      3. div-sub 68.0%

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

   3. Simplified 68.0%

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

   4. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

      2. sub-neg 98.8%

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

      3. mul-1-neg 98.8%

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

      4. *-commutative 98.8%

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

      5. mul-1-neg 98.8%

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

      6. *-commutative 98.8%

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

      7. mul-1-neg 98.8%

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

      8. sub-neg 98.8%

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

      9. mul-1-neg 98.8%

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

      10. +-commutative 98.8%

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

      11. exp-prod 98.8%

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

      12. *-lft-identity 98.8%

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

      13. metadata-eval 98.8%

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

      14. cancel-sign-sub-inv 98.8%

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

      15. exp-prod 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in eps around 0 90.9%

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

   8. Taylor expanded in eps around 0 82.3%

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

      1. exp-neg 82.3%

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

      2. associate-*r/ 82.3%

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

      3. metadata-eval 82.3%

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

   10. Simplified 82.3%

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

   if 5e9 < x < 1.5e158

   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 38.8%

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

      1. mul-1-neg 38.8%

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

      2. +-commutative 38.8%

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

      3. *-commutative 38.8%

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

      4. +-commutative 38.8%

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

   6. Simplified 38.8%

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

   7. Taylor expanded in eps around 0 1.7%

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

      1. +-commutative 1.7%

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

      2. neg-mul-1 1.7%

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

      3. associate--l+ 1.7%

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

      4. expm1-def 1.7%

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

   9. Simplified 1.7%

      \[\leadsto \frac{\color{blue}{\frac{x + \mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 1.7%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{expm1}\left(-x\right)\right)\right)}}{\varepsilon}}{2} \]

       2. expm1-udef 1.7%

          \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(-x\right)\right)} - 1}}{\varepsilon}}{2} \]

       3. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)\right)} - 1}{\varepsilon}}{2} \]

       4. sqrt-unprod 22.5%

          \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)\right)} - 1}{\varepsilon}}{2} \]

       5. sqr-neg 22.5%

          \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)\right)} - 1}{\varepsilon}}{2} \]

       6. sqrt-unprod 22.5%

          \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)\right)} - 1}{\varepsilon}}{2} \]

       7. add-sqr-sqrt 22.5%

          \[\leadsto \frac{\frac{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(\color{blue}{x}\right)\right)} - 1}{\varepsilon}}{2} \]

   11. Applied egg-rr 22.5%

       \[\leadsto \frac{\frac{\color{blue}{e^{\mathsf{log1p}\left(x + \mathsf{expm1}\left(x\right)\right)} - 1}}{\varepsilon}}{2} \]
   12. Step-by-step derivation

       1. expm1-def 22.5%

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{expm1}\left(x\right)\right)\right)}}{\varepsilon}}{2} \]

       2. expm1-log1p 22.5%

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

   13. Simplified 22.5%

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

   if 5.0999999999999999e222 < x < 1.04999999999999998e258

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 45.2%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000000:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{\frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5.1 \cdot 10^{+222} \lor \neg \left(x \leq 1.05 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \]

Alternative 7: 77.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{2}{e^{x}}}{2}\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+222} \lor \neg \left(x \leq 4.5 \cdot 10^{+254}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 (exp x)) 2.0)))
   (if (<= x -4.8e-167)
     t_0
     (if (<= x 3.5e+158)
       (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0)
       (if (or (<= x 5e+222) (not (<= x 4.5e+254)))
         t_0
         (/
          (+
           (* (- 1.0 (* x (- 1.0 eps))) (+ 1.0 (/ 1.0 eps)))
           (* (+ 1.0 (* x (- -1.0 eps))) (/ -1.0 eps)))
          2.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (2.0 / exp(x)) / 2.0;
	double tmp;
	if (x <= -4.8e-167) {
		tmp = t_0;
	} else if (x <= 3.5e+158) {
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	} else if ((x <= 5e+222) || !(x <= 4.5e+254)) {
		tmp = t_0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / exp(x)) / 2.0d0
    if (x <= (-4.8d-167)) then
        tmp = t_0
    else if (x <= 3.5d+158) then
        tmp = (exp((x * (eps + (-1.0d0)))) - (-1.0d0)) / 2.0d0
    else if ((x <= 5d+222) .or. (.not. (x <= 4.5d+254))) then
        tmp = t_0
    else
        tmp = (((1.0d0 - (x * (1.0d0 - eps))) * (1.0d0 + (1.0d0 / eps))) + ((1.0d0 + (x * ((-1.0d0) - eps))) * ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (2.0 / Math.exp(x)) / 2.0;
	double tmp;
	if (x <= -4.8e-167) {
		tmp = t_0;
	} else if (x <= 3.5e+158) {
		tmp = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	} else if ((x <= 5e+222) || !(x <= 4.5e+254)) {
		tmp = t_0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (2.0 / math.exp(x)) / 2.0
	tmp = 0
	if x <= -4.8e-167:
		tmp = t_0
	elif x <= 3.5e+158:
		tmp = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	elif (x <= 5e+222) or not (x <= 4.5e+254):
		tmp = t_0
	else:
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(2.0 / exp(x)) / 2.0)
	tmp = 0.0
	if (x <= -4.8e-167)
		tmp = t_0;
	elseif (x <= 3.5e+158)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0);
	elseif ((x <= 5e+222) || !(x <= 4.5e+254))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps))) * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(1.0 + Float64(x * Float64(-1.0 - eps))) * Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (2.0 / exp(x)) / 2.0;
	tmp = 0.0;
	if (x <= -4.8e-167)
		tmp = t_0;
	elseif (x <= 3.5e+158)
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	elseif ((x <= 5e+222) || ~((x <= 4.5e+254)))
		tmp = t_0;
	else
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -4.8e-167], t$95$0, If[LessEqual[x, 3.5e+158], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5e+222], N[Not[LessEqual[x, 4.5e+254]], $MachinePrecision]], t$95$0, N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999986e-167 or 3.5000000000000001e158 < x < 5.00000000000000023e222 or 4.4999999999999998e254 < x

   1. Initial program 82.6%

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

      1. div-sub 82.6%

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

      2. +-rgt-identity 82.6%

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

      3. div-sub 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. mul-1-neg 98.9%

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

      4. *-commutative 98.9%

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

      5. mul-1-neg 98.9%

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

      6. *-commutative 98.9%

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

      7. mul-1-neg 98.9%

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

      8. sub-neg 98.9%

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

      9. mul-1-neg 98.9%

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

      10. +-commutative 98.9%

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

      11. exp-prod 98.9%

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

      12. *-lft-identity 98.9%

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

      13. metadata-eval 98.9%

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

      14. cancel-sign-sub-inv 98.9%

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

      15. exp-prod 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in eps around 0 87.2%

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

   8. Taylor expanded in eps around 0 79.9%

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

      1. exp-neg 79.9%

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

      2. associate-*r/ 79.9%

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

      3. metadata-eval 79.9%

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

   10. Simplified 79.9%

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

   if -4.79999999999999986e-167 < x < 3.5000000000000001e158

   1. Initial program 66.8%

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

      1. div-sub 66.8%

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

      2. +-rgt-identity 66.8%

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

      3. div-sub 66.8%

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

   3. Simplified 66.8%

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

   4. Taylor expanded in eps around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. mul-1-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. mul-1-neg 99.1%

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

      6. *-commutative 99.1%

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

      7. mul-1-neg 99.1%

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

      8. sub-neg 99.1%

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

      9. mul-1-neg 99.1%

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

      10. +-commutative 99.1%

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

      11. exp-prod 99.1%

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

      12. *-lft-identity 99.1%

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

      13. metadata-eval 99.1%

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

      14. cancel-sign-sub-inv 99.1%

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

      15. exp-prod 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in x around 0 74.1%

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

   if 5.00000000000000023e222 < x < 4.4999999999999998e254

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 45.2%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+222} \lor \neg \left(x \leq 4.5 \cdot 10^{+254}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \]

Alternative 8: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+222} \lor \neg \left(x \leq 9 \cdot 10^{+257}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= x 5.2e+222) (not (<= x 9e+257)))
   (/ (/ 2.0 (exp x)) 2.0)
   (/
    (+
     (* (- 1.0 (* x (- 1.0 eps))) (+ 1.0 (/ 1.0 eps)))
     (* (+ 1.0 (* x (- -1.0 eps))) (/ -1.0 eps)))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((x <= 5.2e+222) || !(x <= 9e+257)) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= 5.2d+222) .or. (.not. (x <= 9d+257))) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (((1.0d0 - (x * (1.0d0 - eps))) * (1.0d0 + (1.0d0 / eps))) + ((1.0d0 + (x * ((-1.0d0) - eps))) * ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if ((x <= 5.2e+222) || !(x <= 9e+257)) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (x <= 5.2e+222) or not (x <= 9e+257):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((x <= 5.2e+222) || !(x <= 9e+257))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps))) * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(1.0 + Float64(x * Float64(-1.0 - eps))) * Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 5.2e+222) || ~((x <= 9e+257)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[x, 5.2e+222], N[Not[LessEqual[x, 9e+257]], $MachinePrecision]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2000000000000002e222 or 8.9999999999999999e257 < x

   1. Initial program 73.3%

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

      1. div-sub 73.3%

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

      2. +-rgt-identity 73.3%

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

      3. div-sub 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in eps around inf 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. *-commutative 99.0%

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

      2. sub-neg 99.0%

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

      3. mul-1-neg 99.0%

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

      4. *-commutative 99.0%

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

      5. mul-1-neg 99.0%

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

      6. *-commutative 99.0%

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

      7. mul-1-neg 99.0%

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

      8. sub-neg 99.0%

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

      9. mul-1-neg 99.0%

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

      10. +-commutative 99.0%

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

      11. exp-prod 99.0%

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

      12. *-lft-identity 99.0%

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

      13. metadata-eval 99.0%

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

      14. cancel-sign-sub-inv 99.0%

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

      15. exp-prod 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in eps around 0 85.2%

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

   8. Taylor expanded in eps around 0 74.5%

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

      1. exp-neg 74.5%

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

      2. associate-*r/ 74.5%

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

      3. metadata-eval 74.5%

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

   10. Simplified 74.5%

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

   if 5.2000000000000002e222 < x < 8.9999999999999999e257

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 45.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+222} \lor \neg \left(x \leq 9 \cdot 10^{+257}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \end{array} \]

Alternative 9: 63.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{if}\;x \leq -2050000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+222}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+259}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* x x) -0.5))))
   (if (<= x -2050000.0)
     (/ (/ (* x (* x 0.5)) eps) 2.0)
     (if (<= x 1.4)
       (/ (+ t_0 t_0) 2.0)
       (if (<= x 9.4e+222)
         0.0
         (if (<= x 1e+259)
           (/
            (+
             (* (- 1.0 (* x (- 1.0 eps))) (+ 1.0 (/ 1.0 eps)))
             (* (+ 1.0 (* x (- -1.0 eps))) (/ -1.0 eps)))
            2.0)
           0.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((x * x) * -0.5);
	double tmp;
	if (x <= -2050000.0) {
		tmp = ((x * (x * 0.5)) / eps) / 2.0;
	} else if (x <= 1.4) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 9.4e+222) {
		tmp = 0.0;
	} else if (x <= 1e+259) {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((x * x) * (-0.5d0))
    if (x <= (-2050000.0d0)) then
        tmp = ((x * (x * 0.5d0)) / eps) / 2.0d0
    else if (x <= 1.4d0) then
        tmp = (t_0 + t_0) / 2.0d0
    else if (x <= 9.4d+222) then
        tmp = 0.0d0
    else if (x <= 1d+259) then
        tmp = (((1.0d0 - (x * (1.0d0 - eps))) * (1.0d0 + (1.0d0 / eps))) + ((1.0d0 + (x * ((-1.0d0) - eps))) * ((-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 t_0 = 1.0 + ((x * x) * -0.5);
	double tmp;
	if (x <= -2050000.0) {
		tmp = ((x * (x * 0.5)) / eps) / 2.0;
	} else if (x <= 1.4) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 9.4e+222) {
		tmp = 0.0;
	} else if (x <= 1e+259) {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((x * x) * -0.5)
	tmp = 0
	if x <= -2050000.0:
		tmp = ((x * (x * 0.5)) / eps) / 2.0
	elif x <= 1.4:
		tmp = (t_0 + t_0) / 2.0
	elif x <= 9.4e+222:
		tmp = 0.0
	elif x <= 1e+259:
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-1.0 / eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(x * x) * -0.5))
	tmp = 0.0
	if (x <= -2050000.0)
		tmp = Float64(Float64(Float64(x * Float64(x * 0.5)) / eps) / 2.0);
	elseif (x <= 1.4)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	elseif (x <= 9.4e+222)
		tmp = 0.0;
	elseif (x <= 1e+259)
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps))) * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(1.0 + Float64(x * Float64(-1.0 - eps))) * Float64(-1.0 / eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 + ((x * x) * -0.5);
	tmp = 0.0;
	if (x <= -2050000.0)
		tmp = ((x * (x * 0.5)) / eps) / 2.0;
	elseif (x <= 1.4)
		tmp = (t_0 + t_0) / 2.0;
	elseif (x <= 9.4e+222)
		tmp = 0.0;
	elseif (x <= 1e+259)
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((1.0 + (x * (-1.0 - eps))) * (-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_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2050000.0], N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.4e+222], 0.0, If[LessEqual[x, 1e+259], N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.05e6

   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 34.1%

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

      1. mul-1-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. *-commutative 34.1%

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

      4. +-commutative 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in eps around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. neg-mul-1 55.3%

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

      3. associate--l+ 55.3%

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

      4. expm1-def 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in x around 0 35.0%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot {x}^{2}}}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 35.0%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot 0.5}}{\varepsilon}}{2} \]

       2. unpow2 35.0%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.5}{\varepsilon}}{2} \]

       3. associate-*l* 35.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   12. Simplified 35.0%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   if -2.05e6 < x < 1.3999999999999999

   1. Initial program 52.9%

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

      1. div-sub 52.9%

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

      2. +-rgt-identity 52.9%

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

      3. div-sub 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in eps around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. distribute-lft1-in 79.9%

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

      3. mul-1-neg 79.9%

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

      4. distribute-lft-out 79.9%

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

      5. mul-1-neg 79.9%

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

      6. *-commutative 79.9%

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

      7. distribute-lft1-in 80.6%

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

      8. mul-1-neg 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around 0 80.1%

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

      1. *-commutative 80.1%

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

      2. unpow2 80.1%

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

   9. Simplified 80.1%

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

   10. Taylor expanded in x around 0 79.4%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

   12. Simplified 79.4%

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

   if 1.3999999999999999 < x < 9.3999999999999998e222 or 9.999999999999999e258 < 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 x around 0 31.0%

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

      1. mul-1-neg 31.0%

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

      2. +-commutative 31.0%

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

      3. *-commutative 31.0%

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

      4. +-commutative 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in x around 0 18.0%

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

   8. Taylor expanded in eps around 0 51.5%

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

      1. distribute-lft1-in 51.5%

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

      2. metadata-eval 51.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

      3. mul0-lft 51.5%

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

      4. +-inverses 51.5%

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

      5. div-sub 22.5%

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

      6. +-inverses 51.5%

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

   10. Simplified 51.5%

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

   if 9.3999999999999998e222 < x < 9.999999999999999e258

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 45.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2050000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(1 + \left(x \cdot x\right) \cdot -0.5\right)}{2}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+222}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+259}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right) \cdot \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 63.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{if}\;x \leq -750000000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+223}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+253}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \frac{x + -1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* x x) -0.5))))
   (if (<= x -750000000.0)
     (/ (/ (* x (* x 0.5)) eps) 2.0)
     (if (<= x 1.4)
       (/ (+ t_0 t_0) 2.0)
       (if (<= x 2.05e+223)
         0.0
         (if (<= x 1.5e+253)
           (/
            (+
             (* (- 1.0 (* x (- 1.0 eps))) (+ 1.0 (/ 1.0 eps)))
             (/ (+ x -1.0) eps))
            2.0)
           0.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((x * x) * -0.5);
	double tmp;
	if (x <= -750000000.0) {
		tmp = ((x * (x * 0.5)) / eps) / 2.0;
	} else if (x <= 1.4) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 2.05e+223) {
		tmp = 0.0;
	} else if (x <= 1.5e+253) {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((x + -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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((x * x) * (-0.5d0))
    if (x <= (-750000000.0d0)) then
        tmp = ((x * (x * 0.5d0)) / eps) / 2.0d0
    else if (x <= 1.4d0) then
        tmp = (t_0 + t_0) / 2.0d0
    else if (x <= 2.05d+223) then
        tmp = 0.0d0
    else if (x <= 1.5d+253) then
        tmp = (((1.0d0 - (x * (1.0d0 - eps))) * (1.0d0 + (1.0d0 / eps))) + ((x + (-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 t_0 = 1.0 + ((x * x) * -0.5);
	double tmp;
	if (x <= -750000000.0) {
		tmp = ((x * (x * 0.5)) / eps) / 2.0;
	} else if (x <= 1.4) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 2.05e+223) {
		tmp = 0.0;
	} else if (x <= 1.5e+253) {
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((x + -1.0) / eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((x * x) * -0.5)
	tmp = 0
	if x <= -750000000.0:
		tmp = ((x * (x * 0.5)) / eps) / 2.0
	elif x <= 1.4:
		tmp = (t_0 + t_0) / 2.0
	elif x <= 2.05e+223:
		tmp = 0.0
	elif x <= 1.5e+253:
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((x + -1.0) / eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(x * x) * -0.5))
	tmp = 0.0
	if (x <= -750000000.0)
		tmp = Float64(Float64(Float64(x * Float64(x * 0.5)) / eps) / 2.0);
	elseif (x <= 1.4)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	elseif (x <= 2.05e+223)
		tmp = 0.0;
	elseif (x <= 1.5e+253)
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps))) * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(x + -1.0) / eps)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 + ((x * x) * -0.5);
	tmp = 0.0;
	if (x <= -750000000.0)
		tmp = ((x * (x * 0.5)) / eps) / 2.0;
	elseif (x <= 1.4)
		tmp = (t_0 + t_0) / 2.0;
	elseif (x <= 2.05e+223)
		tmp = 0.0;
	elseif (x <= 1.5e+253)
		tmp = (((1.0 - (x * (1.0 - eps))) * (1.0 + (1.0 / eps))) + ((x + -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_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -750000000.0], N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.05e+223], 0.0, If[LessEqual[x, 1.5e+253], N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.5e8

   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 34.1%

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

      1. mul-1-neg 34.1%

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

      2. +-commutative 34.1%

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

      3. *-commutative 34.1%

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

      4. +-commutative 34.1%

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

   6. Simplified 34.1%

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

   7. Taylor expanded in eps around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. neg-mul-1 55.3%

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

      3. associate--l+ 55.3%

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

      4. expm1-def 55.3%

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

   9. Simplified 55.3%

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

   10. Taylor expanded in x around 0 35.0%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot {x}^{2}}}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 35.0%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot 0.5}}{\varepsilon}}{2} \]

       2. unpow2 35.0%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.5}{\varepsilon}}{2} \]

       3. associate-*l* 35.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   12. Simplified 35.0%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   if -7.5e8 < x < 1.3999999999999999

   1. Initial program 52.9%

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

      1. div-sub 52.9%

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

      2. +-rgt-identity 52.9%

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

      3. div-sub 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in eps around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. distribute-lft1-in 79.9%

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

      3. mul-1-neg 79.9%

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

      4. distribute-lft-out 79.9%

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

      5. mul-1-neg 79.9%

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

      6. *-commutative 79.9%

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

      7. distribute-lft1-in 80.6%

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

      8. mul-1-neg 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around 0 80.1%

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

      1. *-commutative 80.1%

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

      2. unpow2 80.1%

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

   9. Simplified 80.1%

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

   10. Taylor expanded in x around 0 79.4%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

   12. Simplified 79.4%

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

   if 1.3999999999999999 < x < 2.05e223 or 1.4999999999999999e253 < 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 x around 0 31.0%

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

      1. mul-1-neg 31.0%

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

      2. +-commutative 31.0%

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

      3. *-commutative 31.0%

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

      4. +-commutative 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in x around 0 18.0%

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

   8. Taylor expanded in eps around 0 51.5%

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

      1. distribute-lft1-in 51.5%

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

      2. metadata-eval 51.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

      3. mul0-lft 51.5%

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

      4. +-inverses 51.5%

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

      5. div-sub 22.5%

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

      6. +-inverses 51.5%

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

   10. Simplified 51.5%

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

   if 2.05e223 < x < 1.4999999999999999e253

   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 55.9%

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

      1. mul-1-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. *-commutative 55.9%

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

      4. +-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in x around 0 22.2%

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

   8. Taylor expanded in eps around 0 34.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750000000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(1 + \left(x \cdot x\right) \cdot -0.5\right)}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+223}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+253}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \frac{x + -1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 63.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := 1 + \left(x \cdot x\right) \cdot -0.5\\ t_1 := \frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{if}\;x \leq -750000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+222}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+259}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (* x x) -0.5))) (t_1 (/ (/ (* x (* x 0.5)) eps) 2.0)))
   (if (<= x -750000000.0)
     t_1
     (if (<= x 1.4)
       (/ (+ t_0 t_0) 2.0)
       (if (<= x 5e+222) 0.0 (if (<= x 8.2e+259) t_1 0.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = 1.0 + ((x * x) * -0.5);
	double t_1 = ((x * (x * 0.5)) / eps) / 2.0;
	double tmp;
	if (x <= -750000000.0) {
		tmp = t_1;
	} else if (x <= 1.4) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 5e+222) {
		tmp = 0.0;
	} else if (x <= 8.2e+259) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((x * x) * (-0.5d0))
    t_1 = ((x * (x * 0.5d0)) / eps) / 2.0d0
    if (x <= (-750000000.0d0)) then
        tmp = t_1
    else if (x <= 1.4d0) then
        tmp = (t_0 + t_0) / 2.0d0
    else if (x <= 5d+222) then
        tmp = 0.0d0
    else if (x <= 8.2d+259) then
        tmp = t_1
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = 1.0 + ((x * x) * -0.5);
	double t_1 = ((x * (x * 0.5)) / eps) / 2.0;
	double tmp;
	if (x <= -750000000.0) {
		tmp = t_1;
	} else if (x <= 1.4) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 5e+222) {
		tmp = 0.0;
	} else if (x <= 8.2e+259) {
		tmp = t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = 1.0 + ((x * x) * -0.5)
	t_1 = ((x * (x * 0.5)) / eps) / 2.0
	tmp = 0
	if x <= -750000000.0:
		tmp = t_1
	elif x <= 1.4:
		tmp = (t_0 + t_0) / 2.0
	elif x <= 5e+222:
		tmp = 0.0
	elif x <= 8.2e+259:
		tmp = t_1
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(1.0 + Float64(Float64(x * x) * -0.5))
	t_1 = Float64(Float64(Float64(x * Float64(x * 0.5)) / eps) / 2.0)
	tmp = 0.0
	if (x <= -750000000.0)
		tmp = t_1;
	elseif (x <= 1.4)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	elseif (x <= 5e+222)
		tmp = 0.0;
	elseif (x <= 8.2e+259)
		tmp = t_1;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = 1.0 + ((x * x) * -0.5);
	t_1 = ((x * (x * 0.5)) / eps) / 2.0;
	tmp = 0.0;
	if (x <= -750000000.0)
		tmp = t_1;
	elseif (x <= 1.4)
		tmp = (t_0 + t_0) / 2.0;
	elseif (x <= 5e+222)
		tmp = 0.0;
	elseif (x <= 8.2e+259)
		tmp = t_1;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -750000000.0], t$95$1, If[LessEqual[x, 1.4], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+222], 0.0, If[LessEqual[x, 8.2e+259], t$95$1, 0.0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e8 or 5.00000000000000023e222 < x < 8.2000000000000005e259

   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 38.3%

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

      1. mul-1-neg 38.3%

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

      2. +-commutative 38.3%

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

      3. *-commutative 38.3%

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

      4. +-commutative 38.3%

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

   6. Simplified 38.3%

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

   7. Taylor expanded in eps around 0 45.1%

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

      1. +-commutative 45.1%

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

      2. neg-mul-1 45.1%

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

      3. associate--l+ 45.1%

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

      4. expm1-def 45.1%

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

   9. Simplified 45.1%

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

   10. Taylor expanded in x around 0 34.7%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot {x}^{2}}}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 34.7%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot 0.5}}{\varepsilon}}{2} \]

       2. unpow2 34.7%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.5}{\varepsilon}}{2} \]

       3. associate-*l* 34.7%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   12. Simplified 34.7%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   if -7.5e8 < x < 1.3999999999999999

   1. Initial program 52.9%

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

      1. div-sub 52.9%

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

      2. +-rgt-identity 52.9%

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

      3. div-sub 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in eps around 0 79.9%

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

      1. *-commutative 79.9%

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

      2. distribute-lft1-in 79.9%

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

      3. mul-1-neg 79.9%

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

      4. distribute-lft-out 79.9%

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

      5. mul-1-neg 79.9%

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

      6. *-commutative 79.9%

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

      7. distribute-lft1-in 80.6%

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

      8. mul-1-neg 80.6%

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

   6. Simplified 80.6%

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

   7. Taylor expanded in x around 0 80.1%

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

      1. *-commutative 80.1%

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

      2. unpow2 80.1%

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

   9. Simplified 80.1%

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

   10. Taylor expanded in x around 0 79.4%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

   12. Simplified 79.4%

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

   if 1.3999999999999999 < x < 5.00000000000000023e222 or 8.2000000000000005e259 < 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 x around 0 31.0%

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

      1. mul-1-neg 31.0%

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

      2. +-commutative 31.0%

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

      3. *-commutative 31.0%

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

      4. +-commutative 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in x around 0 18.0%

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

   8. Taylor expanded in eps around 0 51.5%

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

      1. distribute-lft1-in 51.5%

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

      2. metadata-eval 51.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

      3. mul0-lft 51.5%

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

      4. +-inverses 51.5%

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

      5. div-sub 22.5%

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

      6. +-inverses 51.5%

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

   10. Simplified 51.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -750000000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(1 + \left(x \cdot x\right) \cdot -0.5\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+222}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 63.8% accurate, 13.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+222}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+256}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (/ (* x (* x 0.5)) eps) 2.0)))
   (if (<= x -9.8e+64)
     t_0
     (if (<= x 500.0) 1.0 (if (<= x 5e+222) 0.0 (if (<= x 4e+256) t_0 0.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = ((x * (x * 0.5)) / eps) / 2.0;
	double tmp;
	if (x <= -9.8e+64) {
		tmp = t_0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else if (x <= 5e+222) {
		tmp = 0.0;
	} else if (x <= 4e+256) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * (x * 0.5d0)) / eps) / 2.0d0
    if (x <= (-9.8d+64)) then
        tmp = t_0
    else if (x <= 500.0d0) then
        tmp = 1.0d0
    else if (x <= 5d+222) then
        tmp = 0.0d0
    else if (x <= 4d+256) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = ((x * (x * 0.5)) / eps) / 2.0;
	double tmp;
	if (x <= -9.8e+64) {
		tmp = t_0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else if (x <= 5e+222) {
		tmp = 0.0;
	} else if (x <= 4e+256) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = ((x * (x * 0.5)) / eps) / 2.0
	tmp = 0
	if x <= -9.8e+64:
		tmp = t_0
	elif x <= 500.0:
		tmp = 1.0
	elif x <= 5e+222:
		tmp = 0.0
	elif x <= 4e+256:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(Float64(x * Float64(x * 0.5)) / eps) / 2.0)
	tmp = 0.0
	if (x <= -9.8e+64)
		tmp = t_0;
	elseif (x <= 500.0)
		tmp = 1.0;
	elseif (x <= 5e+222)
		tmp = 0.0;
	elseif (x <= 4e+256)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = ((x * (x * 0.5)) / eps) / 2.0;
	tmp = 0.0;
	if (x <= -9.8e+64)
		tmp = t_0;
	elseif (x <= 500.0)
		tmp = 1.0;
	elseif (x <= 5e+222)
		tmp = 0.0;
	elseif (x <= 4e+256)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -9.8e+64], t$95$0, If[LessEqual[x, 500.0], 1.0, If[LessEqual[x, 5e+222], 0.0, If[LessEqual[x, 4e+256], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.8000000000000005e64 or 5.00000000000000023e222 < x < 4.0000000000000001e256

   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 39.9%

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

      1. mul-1-neg 39.9%

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

      2. +-commutative 39.9%

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

      3. *-commutative 39.9%

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

      4. +-commutative 39.9%

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

   6. Simplified 39.9%

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

   7. Taylor expanded in eps around 0 40.9%

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

      1. +-commutative 40.9%

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

      2. neg-mul-1 40.9%

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

      3. associate--l+ 40.9%

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

      4. expm1-def 40.9%

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

   9. Simplified 40.9%

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

   10. Taylor expanded in x around 0 38.6%

       \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot {x}^{2}}}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 38.6%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{2} \cdot 0.5}}{\varepsilon}}{2} \]

       2. unpow2 38.6%

          \[\leadsto \frac{\frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.5}{\varepsilon}}{2} \]

       3. associate-*l* 38.6%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   12. Simplified 38.6%

       \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot 0.5\right)}}{\varepsilon}}{2} \]

   if -9.8000000000000005e64 < x < 500

   1. Initial program 54.5%

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

      1. div-sub 54.5%

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

      2. +-rgt-identity 54.5%

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

      3. div-sub 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in x around 0 76.5%

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

   if 500 < x < 5.00000000000000023e222 or 4.0000000000000001e256 < 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 x around 0 31.0%

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

      1. mul-1-neg 31.0%

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

      2. +-commutative 31.0%

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

      3. *-commutative 31.0%

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

      4. +-commutative 31.0%

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

   6. Simplified 31.0%

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

   7. Taylor expanded in x around 0 18.0%

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

   8. Taylor expanded in eps around 0 51.5%

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

      1. distribute-lft1-in 51.5%

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

      2. metadata-eval 51.5%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

      3. mul0-lft 51.5%

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

      4. +-inverses 51.5%

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

      5. div-sub 22.5%

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

      6. +-inverses 51.5%

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

   10. Simplified 51.5%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+222}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+256}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot 0.5\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 63.9% accurate, 28.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e-9) (/ (* x (- eps)) 2.0) (if (<= x 550.0) 1.0 0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e-9) {
		tmp = (x * -eps) / 2.0;
	} else 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 <= (-2.5d-9)) then
        tmp = (x * -eps) / 2.0d0
    else 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 <= -2.5e-9) {
		tmp = (x * -eps) / 2.0;
	} else 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 <= -2.5e-9:
		tmp = (x * -eps) / 2.0
	elif 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 <= -2.5e-9)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (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 <= -2.5e-9)
		tmp = (x * -eps) / 2.0;
	elseif (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, -2.5e-9], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e-9

   1. Initial program 97.5%

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

      1. div-sub 97.5%

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

      2. +-rgt-identity 97.5%

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

      3. div-sub 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 34.9%

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

      1. mul-1-neg 34.9%

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

      2. +-commutative 34.9%

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

      3. *-commutative 34.9%

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

      4. +-commutative 34.9%

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

   6. Simplified 34.9%

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

   7. Taylor expanded in x around -inf 22.5%

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

   8. Taylor expanded in eps around inf 22.4%

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

      1. associate-*r* 22.4%

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

      2. mul-1-neg 22.4%

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

   10. Simplified 22.4%

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

   if -2.5000000000000001e-9 < x < 550

   1. Initial program 52.9%

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

      1. div-sub 52.9%

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

      2. +-rgt-identity 52.9%

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

      3. div-sub 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in x around 0 80.3%

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

   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

      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 33.8%

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

      1. mul-1-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. *-commutative 33.8%

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

      4. +-commutative 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in x around 0 18.5%

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

   8. Taylor expanded in eps around 0 48.3%

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

      1. distribute-lft1-in 48.3%

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

      2. metadata-eval 48.3%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

      3. mul0-lft 48.3%

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

      4. +-inverses 48.3%

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

      5. div-sub 22.6%

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

      6. +-inverses 48.3%

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

   10. Simplified 48.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 57.1% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 500.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 500.0) {
		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 <= 500.0d0) 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 <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 500.0)
		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 <= 500.0)
		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, 500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 62.9%

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

      1. div-sub 62.9%

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

      2. +-rgt-identity 62.9%

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

      3. div-sub 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in x around 0 62.9%

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

   if 500 < x

   1. Initial program 100.0%

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

      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 33.8%

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

      1. mul-1-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. *-commutative 33.8%

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

      4. +-commutative 33.8%

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

   6. Simplified 33.8%

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

   7. Taylor expanded in x around 0 18.5%

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

   8. Taylor expanded in eps around 0 48.3%

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

      1. distribute-lft1-in 48.3%

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

      2. metadata-eval 48.3%

         \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

      3. mul0-lft 48.3%

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

      4. +-inverses 48.3%

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

      5. div-sub 22.6%

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

      6. +-inverses 48.3%

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

   10. Simplified 48.3%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 16.5% 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 74.2%

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

   1. div-sub 74.2%

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

   2. +-rgt-identity 74.2%

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

   3. div-sub 74.2%

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

3. Simplified 74.2%

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

4. Taylor expanded in x around 0 40.2%

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

   1. mul-1-neg 40.2%

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

   2. +-commutative 40.2%

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

   3. *-commutative 40.2%

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

   4. +-commutative 40.2%

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

6. Simplified 40.2%

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

7. Taylor expanded in x around 0 23.9%

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

8. Taylor expanded in eps around 0 16.5%

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

   1. distribute-lft1-in 16.5%

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

   2. metadata-eval 16.5%

      \[\leadsto \frac{\frac{\color{blue}{0} \cdot x}{\varepsilon}}{2} \]

   3. mul0-lft 16.5%

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

   4. +-inverses 16.5%

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

   5. div-sub 8.7%

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

   6. +-inverses 16.5%

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

10. Simplified 16.5%

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

11. Final simplification 16.5%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023224 
(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))