NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.3%

Time: 16.1s

Alternatives: 15

Speedup: 2.0×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 6 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.0) (not (<= eps 6e-125)))
   (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- eps)))) 2.0)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.0) || !(eps <= 6e-125)) {
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	} else {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-1.0d0)) .or. (.not. (eps <= 6d-125))) then
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((x * -eps))) / 2.0d0
    else
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.0) || !(eps <= 6e-125)) {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.0) or not (eps <= 6e-125):
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((x * -eps))) / 2.0
	else:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.0) || !(eps <= 6e-125))
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.0) || ~((eps <= 6e-125)))
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	else
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.0], N[Not[LessEqual[eps, 6e-125]], $MachinePrecision]], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1 or 5.99999999999999981e-125 < eps

   1. Initial program 90.3%

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

      1. div-sub 90.3%

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

      2. +-rgt-identity 90.3%

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

      3. div-sub 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in eps around inf 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. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-neg-out 100.0%

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

      3. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -1 < eps < 5.99999999999999981e-125

   1. Initial program 36.1%

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

      1. div-sub 36.1%

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

      2. +-rgt-identity 36.1%

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

      3. div-sub 36.1%

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

   3. Simplified 36.1%

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

   4. Taylor expanded in eps around 0 98.9%

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

      1. *-commutative 98.9%

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

      2. distribute-lft1-in 98.9%

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

      3. mul-1-neg 98.9%

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

      4. distribute-lft-out 98.9%

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

      5. mul-1-neg 98.9%

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

      6. *-commutative 98.9%

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

      7. distribute-lft1-in 100.0%

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

      8. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 6 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (+ (exp (- (pow (cbrt (* x (- 1.0 eps))) 3.0))) (exp (* x (- -1.0 eps))))
  2.0))

C

double code(double x, double eps) {
	return (exp(-pow(cbrt((x * (1.0 - eps))), 3.0)) + exp((x * (-1.0 - eps)))) / 2.0;
}

Java

public static double code(double x, double eps) {
	return (Math.exp(-Math.pow(Math.cbrt((x * (1.0 - eps))), 3.0)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(exp(Float64(-(cbrt(Float64(x * Float64(1.0 - eps))) ^ 3.0))) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Exp[(-N[Power[N[Power[N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision])], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.7%

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

   1. div-sub 71.7%

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

   2. +-rgt-identity 71.7%

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

   3. div-sub 71.7%

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

3. Simplified 71.7%

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

4. Taylor expanded in eps around inf 98.9%

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

   1. neg-mul-1 98.9%

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

   2. *-commutative 98.9%

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

   3. mul-1-neg 98.9%

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

   4. exp-prod 98.9%

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

   5. +-commutative 98.9%

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

   6. *-lft-identity 98.9%

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

   7. metadata-eval 98.9%

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

   8. cancel-sign-sub-inv 98.9%

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

   9. exp-prod 98.9%

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

   10. mul-1-neg 98.9%

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

   11. *-commutative 98.9%

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

   12. sub-neg 98.9%

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

   13. mul-1-neg 98.9%

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

   14. remove-double-neg 98.9%

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

   15. distribute-rgt-neg-in 98.9%

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

   16. +-commutative 98.9%

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

6. Simplified 98.9%

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

   1. add-cube-cbrt 98.9%

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

   2. pow3 98.9%

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

8. Applied egg-rr 98.9%

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

9. Final simplification 98.9%

   \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

Alternative 3: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5000000000000 \lor \neg \left(\varepsilon \leq 6.5 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -5000000000000.0) (not (<= eps 6.5e-139)))
   (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- eps)))) 2.0)
   (/ (+ (exp (- x)) (exp (* x (- -1.0 eps)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -5000000000000.0) || !(eps <= 6.5e-139)) {
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	} else {
		tmp = (exp(-x) + exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-5000000000000.0d0)) .or. (.not. (eps <= 6.5d-139))) then
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((x * -eps))) / 2.0d0
    else
        tmp = (exp(-x) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -5000000000000.0) || !(eps <= 6.5e-139)) {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -5000000000000.0) or not (eps <= 6.5e-139):
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((x * -eps))) / 2.0
	else:
		tmp = (math.exp(-x) + math.exp((x * (-1.0 - eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -5000000000000.0) || !(eps <= 6.5e-139))
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -5000000000000.0) || ~((eps <= 6.5e-139)))
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	else
		tmp = (exp(-x) + exp((x * (-1.0 - eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -5000000000000.0], N[Not[LessEqual[eps, 6.5e-139]], $MachinePrecision]], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -5e12 or 6.5e-139 < eps

   1. Initial program 89.5%

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

      1. div-sub 89.5%

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

      2. +-rgt-identity 89.5%

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

      3. div-sub 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. neg-mul-1 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. exp-prod 99.8%

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

      5. +-commutative 99.8%

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

      6. *-lft-identity 99.8%

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

      7. metadata-eval 99.8%

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

      8. cancel-sign-sub-inv 99.8%

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

      9. exp-prod 99.8%

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

      10. mul-1-neg 99.8%

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

      11. *-commutative 99.8%

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

      12. sub-neg 99.8%

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

      13. mul-1-neg 99.8%

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

      14. remove-double-neg 99.8%

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

      15. distribute-rgt-neg-in 99.8%

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

      16. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-lft-neg-out 99.8%

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

      3. *-commutative 99.8%

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

   9. Simplified 99.8%

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

   if -5e12 < eps < 6.5e-139

   1. Initial program 39.9%

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

      1. div-sub 39.9%

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

      2. +-rgt-identity 39.9%

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

      3. div-sub 39.9%

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

   3. Simplified 39.9%

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

   4. Taylor expanded in eps around inf 97.4%

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

      1. neg-mul-1 97.4%

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

      2. *-commutative 97.4%

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

      3. mul-1-neg 97.4%

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

      4. exp-prod 97.4%

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

      5. +-commutative 97.4%

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

      6. *-lft-identity 97.4%

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

      7. metadata-eval 97.4%

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

      8. cancel-sign-sub-inv 97.4%

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

      9. exp-prod 97.4%

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

      10. mul-1-neg 97.4%

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

      11. *-commutative 97.4%

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

      12. sub-neg 97.4%

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

      13. mul-1-neg 97.4%

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

      14. remove-double-neg 97.4%

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

      15. distribute-rgt-neg-in 97.4%

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

      16. +-commutative 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in eps around 0 97.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -5000000000000 \lor \neg \left(\varepsilon \leq 6.5 \cdot 10^{-139}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 4: 84.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-1 - \varepsilon\right)\\ \mathbf{if}\;x \leq 4.8 \cdot 10^{-176} \lor \neg \left(x \leq 8.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{e^{-x} + e^{t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + t_0\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (- -1.0 eps))))
   (if (or (<= x 4.8e-176) (not (<= x 8.2e+14)))
     (/ (+ (exp (- x)) (exp t_0)) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (+ 1.0 t_0)) 2.0))))

C

double code(double x, double eps) {
	double t_0 = x * (-1.0 - eps);
	double tmp;
	if ((x <= 4.8e-176) || !(x <= 8.2e+14)) {
		tmp = (exp(-x) + exp(t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + (1.0 + t_0)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) - eps)
    if ((x <= 4.8d-176) .or. (.not. (x <= 8.2d+14))) then
        tmp = (exp(-x) + exp(t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + (1.0d0 + t_0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (-1.0 - eps);
	double tmp;
	if ((x <= 4.8e-176) || !(x <= 8.2e+14)) {
		tmp = (Math.exp(-x) + Math.exp(t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + (1.0 + t_0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (-1.0 - eps)
	tmp = 0
	if (x <= 4.8e-176) or not (x <= 8.2e+14):
		tmp = (math.exp(-x) + math.exp(t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + (1.0 + t_0)) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(-1.0 - eps))
	tmp = 0.0
	if ((x <= 4.8e-176) || !(x <= 8.2e+14))
		tmp = Float64(Float64(exp(Float64(-x)) + exp(t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + Float64(1.0 + t_0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (-1.0 - eps);
	tmp = 0.0;
	if ((x <= 4.8e-176) || ~((x <= 8.2e+14)))
		tmp = (exp(-x) + exp(t_0)) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + (1.0 + t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, 4.8e-176], N[Not[LessEqual[x, 8.2e+14]], $MachinePrecision]], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.80000000000000012e-176 or 8.2e14 < x

   1. Initial program 75.3%

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

      1. div-sub 75.3%

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

      2. +-rgt-identity 75.3%

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

      3. div-sub 75.3%

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

   3. Simplified 75.3%

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

   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. neg-mul-1 99.0%

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

      2. *-commutative 99.0%

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

      3. mul-1-neg 99.0%

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

      4. exp-prod 99.0%

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

      5. +-commutative 99.0%

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

      6. *-lft-identity 99.0%

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

      7. metadata-eval 99.0%

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

      8. cancel-sign-sub-inv 99.0%

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

      9. exp-prod 99.0%

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

      10. mul-1-neg 99.0%

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

      11. *-commutative 99.0%

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

      12. sub-neg 99.0%

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

      13. mul-1-neg 99.0%

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

      14. remove-double-neg 99.0%

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

      15. distribute-rgt-neg-in 99.0%

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

      16. +-commutative 99.0%

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

   6. Simplified 99.0%

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

   7. Taylor expanded in eps around 0 85.8%

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

   if 4.80000000000000012e-176 < x < 8.2e14

   1. Initial program 52.0%

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

      1. div-sub 52.0%

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

      2. +-rgt-identity 52.0%

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

      3. div-sub 52.0%

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

   3. Simplified 52.0%

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

   4. Taylor expanded in eps around inf 98.6%

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

      1. neg-mul-1 98.6%

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

      2. *-commutative 98.6%

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

      3. mul-1-neg 98.6%

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

      4. exp-prod 98.6%

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

      5. +-commutative 98.6%

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

      6. *-lft-identity 98.6%

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

      7. metadata-eval 98.6%

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

      8. cancel-sign-sub-inv 98.6%

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

      9. exp-prod 98.6%

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

      10. mul-1-neg 98.6%

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

      11. *-commutative 98.6%

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

      12. sub-neg 98.6%

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

      13. mul-1-neg 98.6%

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

      14. remove-double-neg 98.6%

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

      15. distribute-rgt-neg-in 98.6%

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

      16. +-commutative 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around 0 92.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{-176} \lor \neg \left(x \leq 8.2 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 5: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

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

Wolfram

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 71.7%

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

   1. div-sub 71.7%

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

   2. +-rgt-identity 71.7%

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

   3. div-sub 71.7%

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

3. Simplified 71.7%

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

4. Taylor expanded in eps around inf 98.9%

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

   1. neg-mul-1 98.9%

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

   2. *-commutative 98.9%

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

   3. mul-1-neg 98.9%

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

   4. exp-prod 98.9%

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

   5. +-commutative 98.9%

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

   6. *-lft-identity 98.9%

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

   7. metadata-eval 98.9%

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

   8. cancel-sign-sub-inv 98.9%

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

   9. exp-prod 98.9%

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

   10. mul-1-neg 98.9%

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

   11. *-commutative 98.9%

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

   12. sub-neg 98.9%

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

   13. mul-1-neg 98.9%

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

   14. remove-double-neg 98.9%

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

   15. distribute-rgt-neg-in 98.9%

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

   16. +-commutative 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 6: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(x + 1\right) \cdot e^{x}}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+276}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (- (* x eps) x))) 2.0)))
   (if (<= x -2.5e+17)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 2e-174)
       (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
       (if (<= x 700.0)
         t_0
         (if (<= x 8.8e+151)
           (/ (+ (+ 1.0 (* (* x x) -0.5)) (* (+ x 1.0) (exp x))) 2.0)
           (if (<= x 5.2e+276) 0.0 t_0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp(((x * eps) - x))) / 2.0;
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2e-174) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if (x <= 700.0) {
		tmp = t_0;
	} else if (x <= 8.8e+151) {
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * exp(x))) / 2.0;
	} else if (x <= 5.2e+276) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + exp(((x * eps) - x))) / 2.0d0
    if (x <= (-2.5d+17)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 2d-174) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if (x <= 700.0d0) then
        tmp = t_0
    else if (x <= 8.8d+151) then
        tmp = ((1.0d0 + ((x * x) * (-0.5d0))) + ((x + 1.0d0) * exp(x))) / 2.0d0
    else if (x <= 5.2d+276) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp(((x * eps) - x))) / 2.0;
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2e-174) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if (x <= 700.0) {
		tmp = t_0;
	} else if (x <= 8.8e+151) {
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * Math.exp(x))) / 2.0;
	} else if (x <= 5.2e+276) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp(((x * eps) - x))) / 2.0
	tmp = 0
	if x <= -2.5e+17:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2e-174:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif x <= 700.0:
		tmp = t_0
	elif x <= 8.8e+151:
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * math.exp(x))) / 2.0
	elif x <= 5.2e+276:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(Float64(Float64(x * eps) - x))) / 2.0)
	tmp = 0.0
	if (x <= -2.5e+17)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2e-174)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif (x <= 700.0)
		tmp = t_0;
	elseif (x <= 8.8e+151)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -0.5)) + Float64(Float64(x + 1.0) * exp(x))) / 2.0);
	elseif (x <= 5.2e+276)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + exp(((x * eps) - x))) / 2.0;
	tmp = 0.0;
	if (x <= -2.5e+17)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 2e-174)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif (x <= 700.0)
		tmp = t_0;
	elseif (x <= 8.8e+151)
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * exp(x))) / 2.0;
	elseif (x <= 5.2e+276)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.5e+17], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e-174], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 700.0], t$95$0, If[LessEqual[x, 8.8e+151], N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.2e+276], 0.0, t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.5e17

   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. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{1 + e^{-{1}^{0.3333333333333333} \cdot x}}}{2} \]
   13. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   14. Simplified 100.0%

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

   if -2.5e17 < x < 2e-174

   1. Initial program 48.2%

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

      1. div-sub 48.2%

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

      2. +-rgt-identity 48.2%

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

      3. div-sub 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in eps around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. *-commutative 97.8%

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

      3. mul-1-neg 97.8%

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

      4. exp-prod 97.8%

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

      5. +-commutative 97.8%

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

      6. *-lft-identity 97.8%

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

      7. metadata-eval 97.8%

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

      8. cancel-sign-sub-inv 97.8%

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

      9. exp-prod 97.8%

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

      10. mul-1-neg 97.8%

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

      11. *-commutative 97.8%

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

      12. sub-neg 97.8%

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

      13. mul-1-neg 97.8%

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

      14. remove-double-neg 97.8%

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

      15. distribute-rgt-neg-in 97.8%

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

      16. +-commutative 97.8%

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

   6. Simplified 97.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in eps around inf 97.9%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 97.9%

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

       2. distribute-lft-neg-out 97.9%

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

       3. *-commutative 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in x around inf 90.7%

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

       1. associate-*r* 90.7%

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

       2. mul-1-neg 90.7%

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

   14. Simplified 90.7%

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

   if 2e-174 < x < 700 or 5.19999999999999998e276 < x

   1. Initial program 59.1%

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

      1. div-sub 59.1%

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

      2. +-rgt-identity 59.1%

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

      3. div-sub 59.1%

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

   3. Simplified 59.1%

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

   4. Taylor expanded in eps around inf 98.8%

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

      1. neg-mul-1 98.8%

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

      2. *-commutative 98.8%

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

      3. mul-1-neg 98.8%

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

      4. exp-prod 98.8%

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

      5. +-commutative 98.8%

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

      6. *-lft-identity 98.8%

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

      7. metadata-eval 98.8%

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

      8. cancel-sign-sub-inv 98.8%

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

      9. exp-prod 98.8%

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

      10. mul-1-neg 98.8%

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

      11. *-commutative 98.8%

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

      12. sub-neg 98.8%

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

      13. mul-1-neg 98.8%

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

      14. remove-double-neg 98.8%

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

      15. distribute-rgt-neg-in 98.8%

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

      16. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in x around 0 85.2%

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

   8. Taylor expanded in x around inf 85.2%

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

   if 700 < x < 8.80000000000000027e151

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 30.4%

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

      1. *-commutative 30.4%

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

      2. distribute-lft1-in 30.4%

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

      3. mul-1-neg 30.4%

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

      4. distribute-lft-out 30.4%

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

      5. mul-1-neg 30.4%

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

      6. *-commutative 30.4%

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

      7. distribute-lft1-in 30.4%

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

      8. mul-1-neg 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

      2. unpow2 0.9%

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

   9. Simplified 0.9%

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

       1. *-commutative 0.9%

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

       2. +-commutative 0.9%

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

       3. distribute-lft-in 0.9%

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

       4. *-commutative 0.9%

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

       5. *-un-lft-identity 0.9%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 71.2%

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

       8. sqr-neg 71.2%

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

       9. sqrt-unprod 71.2%

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

       10. add-sqr-sqrt 71.2%

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

       11. add-sqr-sqrt 0.0%

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

       12. sqrt-unprod 71.2%

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

       13. sqr-neg 71.2%

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

       14. sqrt-unprod 71.2%

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

       15. add-sqr-sqrt 71.2%

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

   11. Applied egg-rr 71.2%

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

       1. *-commutative 71.2%

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

       2. distribute-rgt1-in 71.2%

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

       3. +-commutative 71.2%

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

   13. Simplified 71.2%

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

   if 8.80000000000000027e151 < x < 5.19999999999999998e276

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 53.9%

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

      1. div-sub 53.9%

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

      2. rec-exp 53.9%

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

      3. mul-1-neg 53.9%

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

      4. +-inverses 53.9%

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

   6. Simplified 53.9%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-174}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(x + 1\right) \cdot e^{x}}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+276}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \end{array} \]

Alternative 7: 77.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-175}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+152}:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(x + 1\right) \cdot e^{x}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+284}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e+17)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 2.15e-175)
     (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
     (if (<= x 700.0)
       (/ (+ (exp (* x (+ eps -1.0))) (+ 1.0 (* x (- -1.0 eps)))) 2.0)
       (if (<= x 1e+152)
         (/ (+ (+ 1.0 (* (* x x) -0.5)) (* (+ x 1.0) (exp x))) 2.0)
         (if (<= x 2.1e+284) 0.0 (/ (+ 1.0 (exp (- (* x eps) x))) 2.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2.15e-175) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if (x <= 700.0) {
		tmp = (exp((x * (eps + -1.0))) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1e+152) {
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * exp(x))) / 2.0;
	} else if (x <= 2.1e+284) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.5d+17)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 2.15d-175) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if (x <= 700.0d0) then
        tmp = (exp((x * (eps + (-1.0d0)))) + (1.0d0 + (x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 1d+152) then
        tmp = ((1.0d0 + ((x * x) * (-0.5d0))) + ((x + 1.0d0) * exp(x))) / 2.0d0
    else if (x <= 2.1d+284) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + exp(((x * eps) - x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2.15e-175) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if (x <= 700.0) {
		tmp = (Math.exp((x * (eps + -1.0))) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1e+152) {
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * Math.exp(x))) / 2.0;
	} else if (x <= 2.1e+284) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + Math.exp(((x * eps) - x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.5e+17:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2.15e-175:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif x <= 700.0:
		tmp = (math.exp((x * (eps + -1.0))) + (1.0 + (x * (-1.0 - eps)))) / 2.0
	elif x <= 1e+152:
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * math.exp(x))) / 2.0
	elif x <= 2.1e+284:
		tmp = 0.0
	else:
		tmp = (1.0 + math.exp(((x * eps) - x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e+17)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2.15e-175)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif (x <= 700.0)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + Float64(1.0 + Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1e+152)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(x * x) * -0.5)) + Float64(Float64(x + 1.0) * exp(x))) / 2.0);
	elseif (x <= 2.1e+284)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(x * eps) - x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.5e+17)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 2.15e-175)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif (x <= 700.0)
		tmp = (exp((x * (eps + -1.0))) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 1e+152)
		tmp = ((1.0 + ((x * x) * -0.5)) + ((x + 1.0) * exp(x))) / 2.0;
	elseif (x <= 2.1e+284)
		tmp = 0.0;
	else
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e+17], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.15e-175], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 700.0], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+152], N[(N[(N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.1e+284], 0.0, N[(N[(1.0 + N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -2.5e17

   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. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{1 + e^{-{1}^{0.3333333333333333} \cdot x}}}{2} \]
   13. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   14. Simplified 100.0%

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

   if -2.5e17 < x < 2.14999999999999999e-175

   1. Initial program 48.2%

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

      1. div-sub 48.2%

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

      2. +-rgt-identity 48.2%

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

      3. div-sub 48.2%

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

   3. Simplified 48.2%

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

   4. Taylor expanded in eps around inf 97.8%

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

      1. neg-mul-1 97.8%

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

      2. *-commutative 97.8%

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

      3. mul-1-neg 97.8%

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

      4. exp-prod 97.8%

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

      5. +-commutative 97.8%

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

      6. *-lft-identity 97.8%

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

      7. metadata-eval 97.8%

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

      8. cancel-sign-sub-inv 97.8%

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

      9. exp-prod 97.8%

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

      10. mul-1-neg 97.8%

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

      11. *-commutative 97.8%

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

      12. sub-neg 97.8%

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

      13. mul-1-neg 97.8%

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

      14. remove-double-neg 97.8%

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

      15. distribute-rgt-neg-in 97.8%

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

      16. +-commutative 97.8%

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

   6. Simplified 97.8%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in eps around inf 97.9%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 97.9%

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

       2. distribute-lft-neg-out 97.9%

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

       3. *-commutative 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in x around inf 90.7%

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

       1. associate-*r* 90.7%

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

       2. mul-1-neg 90.7%

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

   14. Simplified 90.7%

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

   if 2.14999999999999999e-175 < x < 700

   1. Initial program 46.6%

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

      1. div-sub 46.6%

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

      2. +-rgt-identity 46.6%

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

      3. div-sub 46.6%

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

   3. Simplified 46.6%

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

   4. Taylor expanded in eps around inf 98.5%

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

      1. neg-mul-1 98.5%

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

      2. *-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. exp-prod 98.5%

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

      5. +-commutative 98.5%

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

      6. *-lft-identity 98.5%

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

      7. metadata-eval 98.5%

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

      8. cancel-sign-sub-inv 98.5%

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

      9. exp-prod 98.5%

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

      10. mul-1-neg 98.5%

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

      11. *-commutative 98.5%

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

      12. sub-neg 98.5%

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

      13. mul-1-neg 98.5%

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

      14. remove-double-neg 98.5%

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

      15. distribute-rgt-neg-in 98.5%

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

      16. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in x around 0 91.9%

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

   if 700 < x < 1e152

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 30.4%

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

      1. *-commutative 30.4%

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

      2. distribute-lft1-in 30.4%

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

      3. mul-1-neg 30.4%

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

      4. distribute-lft-out 30.4%

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

      5. mul-1-neg 30.4%

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

      6. *-commutative 30.4%

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

      7. distribute-lft1-in 30.4%

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

      8. mul-1-neg 30.4%

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

   6. Simplified 30.4%

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

   7. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

      2. unpow2 0.9%

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

   9. Simplified 0.9%

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

       1. *-commutative 0.9%

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

       2. +-commutative 0.9%

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

       3. distribute-lft-in 0.9%

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

       4. *-commutative 0.9%

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

       5. *-un-lft-identity 0.9%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 71.2%

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

       8. sqr-neg 71.2%

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

       9. sqrt-unprod 71.2%

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

       10. add-sqr-sqrt 71.2%

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

       11. add-sqr-sqrt 0.0%

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

       12. sqrt-unprod 71.2%

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

       13. sqr-neg 71.2%

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

       14. sqrt-unprod 71.2%

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

       15. add-sqr-sqrt 71.2%

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

   11. Applied egg-rr 71.2%

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

       1. *-commutative 71.2%

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

       2. distribute-rgt1-in 71.2%

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

       3. +-commutative 71.2%

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

   13. Simplified 71.2%

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

   if 1e152 < x < 2.10000000000000005e284

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 53.9%

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

      1. div-sub 53.9%

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

      2. rec-exp 53.9%

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

      3. mul-1-neg 53.9%

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

      4. +-inverses 53.9%

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

   6. Simplified 53.9%

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

   if 2.10000000000000005e284 < 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. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 64.8%

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

   8. Taylor expanded in x around inf 64.8%

      \[\leadsto \frac{\color{blue}{1 + e^{\varepsilon \cdot x - x}}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-175}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+152}:\\ \;\;\;\;\frac{\left(1 + \left(x \cdot x\right) \cdot -0.5\right) + \left(x + 1\right) \cdot e^{x}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+284}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \end{array} \]

Alternative 8: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ t_1 := \frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+284}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (- (* x eps) x))) 2.0))
        (t_1 (/ (+ 1.0 (exp (* x (- eps)))) 2.0)))
   (if (<= x -2.5e+17)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 5e-176)
       t_1
       (if (<= x 4.2e+69)
         t_0
         (if (<= x 1.76e+158) t_1 (if (<= x 3.5e+284) 0.0 t_0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp(((x * eps) - x))) / 2.0;
	double t_1 = (1.0 + exp((x * -eps))) / 2.0;
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 5e-176) {
		tmp = t_1;
	} else if (x <= 4.2e+69) {
		tmp = t_0;
	} else if (x <= 1.76e+158) {
		tmp = t_1;
	} else if (x <= 3.5e+284) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + exp(((x * eps) - x))) / 2.0d0
    t_1 = (1.0d0 + exp((x * -eps))) / 2.0d0
    if (x <= (-2.5d+17)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 5d-176) then
        tmp = t_1
    else if (x <= 4.2d+69) then
        tmp = t_0
    else if (x <= 1.76d+158) then
        tmp = t_1
    else if (x <= 3.5d+284) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp(((x * eps) - x))) / 2.0;
	double t_1 = (1.0 + Math.exp((x * -eps))) / 2.0;
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 5e-176) {
		tmp = t_1;
	} else if (x <= 4.2e+69) {
		tmp = t_0;
	} else if (x <= 1.76e+158) {
		tmp = t_1;
	} else if (x <= 3.5e+284) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp(((x * eps) - x))) / 2.0
	t_1 = (1.0 + math.exp((x * -eps))) / 2.0
	tmp = 0
	if x <= -2.5e+17:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 5e-176:
		tmp = t_1
	elif x <= 4.2e+69:
		tmp = t_0
	elif x <= 1.76e+158:
		tmp = t_1
	elif x <= 3.5e+284:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(Float64(Float64(x * eps) - x))) / 2.0)
	t_1 = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0)
	tmp = 0.0
	if (x <= -2.5e+17)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 5e-176)
		tmp = t_1;
	elseif (x <= 4.2e+69)
		tmp = t_0;
	elseif (x <= 1.76e+158)
		tmp = t_1;
	elseif (x <= 3.5e+284)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + exp(((x * eps) - x))) / 2.0;
	t_1 = (1.0 + exp((x * -eps))) / 2.0;
	tmp = 0.0;
	if (x <= -2.5e+17)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 5e-176)
		tmp = t_1;
	elseif (x <= 4.2e+69)
		tmp = t_0;
	elseif (x <= 1.76e+158)
		tmp = t_1;
	elseif (x <= 3.5e+284)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.5e+17], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e-176], t$95$1, If[LessEqual[x, 4.2e+69], t$95$0, If[LessEqual[x, 1.76e+158], t$95$1, If[LessEqual[x, 3.5e+284], 0.0, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.5e17

   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. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{1 + e^{-{1}^{0.3333333333333333} \cdot x}}}{2} \]
   13. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   14. Simplified 100.0%

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

   if -2.5e17 < x < 5e-176 or 4.2000000000000003e69 < x < 1.7600000000000001e158

   1. Initial program 58.0%

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

      1. div-sub 58.0%

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

      2. +-rgt-identity 58.0%

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

      3. div-sub 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in eps around inf 98.3%

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

      1. neg-mul-1 98.3%

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

      2. *-commutative 98.3%

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

      3. mul-1-neg 98.3%

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

      4. exp-prod 98.3%

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

      5. +-commutative 98.3%

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

      6. *-lft-identity 98.3%

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

      7. metadata-eval 98.3%

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

      8. cancel-sign-sub-inv 98.3%

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

      9. exp-prod 98.3%

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

      10. mul-1-neg 98.3%

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

      11. *-commutative 98.3%

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

      12. sub-neg 98.3%

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

      13. mul-1-neg 98.3%

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

      14. remove-double-neg 98.3%

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

      15. distribute-rgt-neg-in 98.3%

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

      16. +-commutative 98.3%

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

   6. Simplified 98.3%

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

      1. add-cube-cbrt 98.3%

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

      2. pow3 98.3%

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

   8. Applied egg-rr 98.3%

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

   9. Taylor expanded in eps around inf 93.0%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 92.9%

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

       2. distribute-lft-neg-out 92.9%

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

       3. *-commutative 92.9%

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

   11. Simplified 93.0%

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

   12. Taylor expanded in x around inf 84.1%

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

       1. associate-*r* 84.1%

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

       2. mul-1-neg 84.1%

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

   14. Simplified 84.1%

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

   if 5e-176 < x < 4.2000000000000003e69 or 3.50000000000000019e284 < x

   1. Initial program 71.3%

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

      1. div-sub 71.3%

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

      2. +-rgt-identity 71.3%

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

      3. div-sub 71.3%

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

   3. Simplified 71.3%

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

   4. Taylor expanded in eps around inf 99.2%

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

      1. neg-mul-1 99.2%

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

      2. *-commutative 99.2%

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

      3. mul-1-neg 99.2%

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

      4. exp-prod 99.2%

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

      5. +-commutative 99.2%

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

      6. *-lft-identity 99.2%

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

      7. metadata-eval 99.2%

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

      8. cancel-sign-sub-inv 99.2%

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

      9. exp-prod 99.2%

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

      10. mul-1-neg 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. mul-1-neg 99.2%

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

      14. remove-double-neg 99.2%

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

      15. distribute-rgt-neg-in 99.2%

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

      16. +-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around 0 73.7%

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

   8. Taylor expanded in x around inf 73.7%

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

   if 1.7600000000000001e158 < x < 3.50000000000000019e284

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.9%

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

      1. div-sub 55.9%

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

      2. rec-exp 55.9%

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

      3. mul-1-neg 55.9%

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

      4. +-inverses 55.9%

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

   6. Simplified 55.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-176}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+284}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \end{array} \]

Alternative 9: 74.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e+17)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 2.15e+158) (/ (+ 1.0 (exp (* x (- eps)))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2.15e+158) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.5d+17)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 2.15d+158) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e+17) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2.15e+158) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.5e+17:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2.15e+158:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e+17)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2.15e+158)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.5e+17)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 2.15e+158)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e+17], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.15e+158], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5e17

   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. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-lft-identity 100.0%

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

      7. metadata-eval 100.0%

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

      8. cancel-sign-sub-inv 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

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

      2. pow3 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{\color{blue}{1 + e^{-{1}^{0.3333333333333333} \cdot x}}}{2} \]
   13. Step-by-step derivation

       1. pow-base-1 100.0%

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

       2. *-lft-identity 100.0%

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

   14. Simplified 100.0%

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

   if -2.5e17 < x < 2.15e158

   1. Initial program 60.4%

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

      1. div-sub 60.4%

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

      2. +-rgt-identity 60.4%

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

      3. div-sub 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in eps around inf 98.5%

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

      1. neg-mul-1 98.5%

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

      2. *-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. exp-prod 98.5%

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

      5. +-commutative 98.5%

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

      6. *-lft-identity 98.5%

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

      7. metadata-eval 98.5%

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

      8. cancel-sign-sub-inv 98.5%

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

      9. exp-prod 98.5%

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

      10. mul-1-neg 98.5%

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

      11. *-commutative 98.5%

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

      12. sub-neg 98.5%

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

      13. mul-1-neg 98.5%

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

      14. remove-double-neg 98.5%

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

      15. distribute-rgt-neg-in 98.5%

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

      16. +-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.5%

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

   8. Applied egg-rr 98.5%

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

   9. Taylor expanded in eps around inf 91.7%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 91.6%

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

       2. distribute-lft-neg-out 91.6%

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

       3. *-commutative 91.6%

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

   11. Simplified 91.7%

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

   12. Taylor expanded in x around inf 76.4%

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

       1. associate-*r* 76.4%

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

       2. mul-1-neg 76.4%

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

   14. Simplified 76.4%

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

   if 2.15e158 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 48.3%

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

      1. div-sub 48.3%

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

      2. rec-exp 48.3%

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

      3. mul-1-neg 48.3%

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

      4. +-inverses 48.3%

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

   6. Simplified 48.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 70.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.75e+15) (/ (+ 1.0 (exp (- x))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.75e+15) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.75d+15) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.75e+15) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.75e+15:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.75e+15)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.75e+15)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.75e+15], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.75e15

   1. Initial program 59.0%

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

      1. div-sub 59.0%

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

      2. +-rgt-identity 59.0%

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

      3. div-sub 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in eps around inf 98.4%

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

      1. neg-mul-1 98.4%

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

      2. *-commutative 98.4%

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

      3. mul-1-neg 98.4%

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

      4. exp-prod 98.4%

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

      5. +-commutative 98.4%

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

      6. *-lft-identity 98.4%

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

      7. metadata-eval 98.4%

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

      8. cancel-sign-sub-inv 98.4%

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

      9. exp-prod 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. sub-neg 98.4%

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

      13. mul-1-neg 98.4%

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

      14. remove-double-neg 98.4%

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

      15. distribute-rgt-neg-in 98.4%

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

      16. +-commutative 98.4%

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

   6. Simplified 98.4%

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

      1. add-cube-cbrt 98.4%

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

      2. pow3 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in eps around inf 98.5%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 98.5%

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

       2. distribute-lft-neg-out 98.5%

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

       3. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   12. Taylor expanded in eps around 0 77.0%

       \[\leadsto \frac{\color{blue}{1 + e^{-{1}^{0.3333333333333333} \cdot x}}}{2} \]
   13. Step-by-step derivation

       1. pow-base-1 77.0%

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

       2. *-lft-identity 77.0%

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

   14. Simplified 77.0%

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

   if 1.75e15 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 41.4%

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

      1. div-sub 41.4%

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

      2. rec-exp 41.4%

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

      3. mul-1-neg 41.4%

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

      4. +-inverses 41.4%

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

   6. Simplified 41.4%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 59.6% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7.8e+14) (/ (+ 2.0 (* x eps)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.8e+14) {
		tmp = (2.0 + (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.8d+14) then
        tmp = (2.0d0 + (x * eps)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.8e+14) {
		tmp = (2.0 + (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.8e+14:
		tmp = (2.0 + (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.8e+14)
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.8e+14)
		tmp = (2.0 + (x * eps)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.8e+14], N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.8e14

   1. Initial program 58.8%

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

   3. Simplified 45.7%

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

   4. Taylor expanded in x around 0 59.2%

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

   5. Taylor expanded in eps around 0 62.0%

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

   6. Taylor expanded in eps around 0 62.0%

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

   if 7.8e14 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 40.9%

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

      1. div-sub 40.9%

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

      2. rec-exp 40.9%

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

      3. mul-1-neg 40.9%

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

      4. +-inverses 40.9%

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

   6. Simplified 40.9%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 60.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.5) (/ (- 2.0 (* x eps)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.5) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.5:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.5], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 57.6%

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

      1. div-sub 57.6%

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

      2. +-rgt-identity 57.6%

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

      3. div-sub 57.6%

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

   3. Simplified 57.6%

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

   4. Taylor expanded in eps around inf 98.4%

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

      1. neg-mul-1 98.4%

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

      2. *-commutative 98.4%

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

      3. mul-1-neg 98.4%

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

      4. exp-prod 98.4%

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

      5. +-commutative 98.4%

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

      6. *-lft-identity 98.4%

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

      7. metadata-eval 98.4%

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

      8. cancel-sign-sub-inv 98.4%

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

      9. exp-prod 98.4%

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

      10. mul-1-neg 98.4%

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

      11. *-commutative 98.4%

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

      12. sub-neg 98.4%

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

      13. mul-1-neg 98.4%

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

      14. remove-double-neg 98.4%

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

      15. distribute-rgt-neg-in 98.4%

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

      16. +-commutative 98.4%

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

   6. Simplified 98.4%

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

      1. add-cube-cbrt 98.4%

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

      2. pow3 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in eps around inf 98.4%

      \[\leadsto \frac{e^{-{\left(\sqrt[3]{x \cdot \left(1 - \varepsilon\right)}\right)}^{3}} - \left(-e^{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}\right)}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 98.4%

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

       2. distribute-lft-neg-out 98.4%

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

       3. *-commutative 98.4%

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

   11. Simplified 98.4%

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

   12. Taylor expanded in x around 0 65.4%

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

       1. +-commutative 65.4%

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

       2. mul-1-neg 65.4%

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

       3. unsub-neg 65.4%

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

   14. Simplified 65.4%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 38.6%

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

      1. div-sub 38.6%

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

      2. rec-exp 38.6%

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

      3. mul-1-neg 38.6%

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

      4. +-inverses 38.6%

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

   6. Simplified 38.6%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 59.8% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+72}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7e+72) (/ (* x eps) 2.0) (if (<= x 1.75e+15) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7e+72) {
		tmp = (x * eps) / 2.0;
	} else if (x <= 1.75e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-7d+72)) then
        tmp = (x * eps) / 2.0d0
    else if (x <= 1.75d+15) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7e+72) {
		tmp = (x * eps) / 2.0;
	} else if (x <= 1.75e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7e+72:
		tmp = (x * eps) / 2.0
	elif x <= 1.75e+15:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7e+72)
		tmp = Float64(Float64(x * eps) / 2.0);
	elseif (x <= 1.75e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7e+72)
		tmp = (x * eps) / 2.0;
	elseif (x <= 1.75e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7e+72], N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.75e+15], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000002e72

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 3.2%

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

   5. Taylor expanded in eps around 0 23.0%

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

   6. Taylor expanded in eps around inf 23.0%

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

   if -7.0000000000000002e72 < x < 1.75e15

   1. Initial program 51.3%

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

      1. div-sub 51.3%

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

      2. +-rgt-identity 51.3%

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

      3. div-sub 51.3%

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

   3. Simplified 51.3%

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

   4. Taylor expanded in x around 0 69.4%

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

   if 1.75e15 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 41.4%

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

      1. div-sub 41.4%

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

      2. rec-exp 41.4%

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

      3. mul-1-neg 41.4%

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

      4. +-inverses 41.4%

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

   6. Simplified 41.4%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+72}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 57.0% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 1.75e+15) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.75e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.75d+15) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.75e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.75e+15:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.75e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.75e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.75e+15], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.75e15

   1. Initial program 59.0%

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

      1. div-sub 59.0%

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

      2. +-rgt-identity 59.0%

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

      3. div-sub 59.0%

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

   3. Simplified 59.0%

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

   4. Taylor expanded in x around 0 58.9%

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

   if 1.75e15 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 41.4%

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

      1. div-sub 41.4%

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

      2. rec-exp 41.4%

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

      3. mul-1-neg 41.4%

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

      4. +-inverses 41.4%

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

   6. Simplified 41.4%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 16.3% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 71.7%

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

3. Simplified 62.7%

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

4. Taylor expanded in eps around 0 14.3%

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

   1. div-sub 14.3%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   2. rec-exp 14.3%

      \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

   3. mul-1-neg 14.3%

      \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

   4. +-inverses 14.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

6. Simplified 14.5%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Final simplification 14.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023171 
(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))