NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.6%

Time: 24.1s

Alternatives: 16

Speedup: 0.5×

• 39.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(-1 - \frac{-1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (-
          (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
          (* (- -1.0 (/ -1.0 eps)) (exp (* x (- -1.0 eps)))))))
   (if (<= t_0 0.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
     (/ t_0 2.0))))

C

double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - ((-1.0 - (-1.0 / eps)) * exp((x * (-1.0 - eps))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else {
		tmp = 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 = ((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) - (((-1.0d0) - ((-1.0d0) / eps)) * exp((x * ((-1.0d0) - eps))))
    if (t_0 <= 0.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) - ((-1.0 - (-1.0 / eps)) * Math.exp((x * (-1.0 - eps))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) - ((-1.0 - (-1.0 / eps)) * math.exp((x * (-1.0 - eps))))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 0.0

   1. Initial program 33.1%

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

      1. sub-neg 33.1%

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

      2. neg-sub0 33.1%

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

      3. associate-+r- 33.1%

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

   3. Simplified 33.1%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. rec-exp 100.0%

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

      3. *-commutative 100.0%

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

      4. neg-mul-1 100.0%

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

      5. rec-exp 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. rec-exp 100.0%

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

      8. distribute-lft-out 100.0%

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

      9. mul-1-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. rec-exp 100.0%

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

      12. *-commutative 100.0%

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

      13. neg-mul-1 100.0%

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

      14. rec-exp 100.0%

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

      15. distribute-lft1-in 100.0%

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

      16. rec-exp 100.0%

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

   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%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 66.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 40:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+46} \lor \neg \left(x \leq 6.5 \cdot 10^{+88}\right) \land x \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-299)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 40.0)
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (if (or (<= x 1.32e+46) (and (not (<= x 6.5e+88)) (<= x 1.6e+168)))
       (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
       (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-299) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 40.0) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if ((x <= 1.32e+46) || (!(x <= 6.5e+88) && (x <= 1.6e+168))) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-4d-299)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 40.0d0) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if ((x <= 1.32d+46) .or. (.not. (x <= 6.5d+88)) .and. (x <= 1.6d+168)) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4e-299) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 40.0) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if ((x <= 1.32e+46) || (!(x <= 6.5e+88) && (x <= 1.6e+168))) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4e-299:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 40.0:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif (x <= 1.32e+46) or (not (x <= 6.5e+88) and (x <= 1.6e+168)):
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-299)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 40.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif ((x <= 1.32e+46) || (!(x <= 6.5e+88) && (x <= 1.6e+168)))
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4e-299)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 40.0)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif ((x <= 1.32e+46) || (~((x <= 6.5e+88)) && (x <= 1.6e+168)))
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-299], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 40.0], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.32e+46], And[N[Not[LessEqual[x, 6.5e+88]], $MachinePrecision], LessEqual[x, 1.6e+168]]], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.99999999999999997e-299

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. neg-sub0 61.8%

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

      3. associate-+r- 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 42.4%

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

   5. Taylor expanded in eps around inf 79.7%

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

      1. mul-1-neg 79.7%

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

      2. exp-prod 79.7%

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

      3. +-commutative 79.7%

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

      4. remove-double-neg 79.7%

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

      5. sub-neg 79.7%

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

      6. neg-mul-1 79.7%

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

      7. exp-prod 79.7%

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

      8. mul-1-neg 79.7%

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

      9. *-commutative 79.7%

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

      10. sub-neg 79.7%

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

      11. neg-mul-1 79.7%

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

      12. remove-double-neg 79.7%

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

   7. Simplified 79.7%

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

   if -3.99999999999999997e-299 < x < 40

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. neg-sub0 57.9%

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

      3. associate-+r- 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in x around 0 41.3%

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

   5. Taylor expanded in eps around inf 83.4%

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

      1. exp-prod 83.4%

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

      2. *-commutative 83.4%

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

      3. sub-neg 83.4%

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

      4. neg-mul-1 83.4%

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

      5. *-commutative 83.4%

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

      6. exp-prod 83.4%

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

      7. +-commutative 83.4%

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

      8. associate-*r* 83.4%

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

      9. neg-mul-1 83.4%

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

      10. sub-neg 83.4%

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

      11. *-commutative 83.4%

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

      12. neg-mul-1 83.4%

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

   7. Simplified 83.4%

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

   8. Taylor expanded in eps around inf 83.4%

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

   if 40 < x < 1.32e46 or 6.5000000000000002e88 < x < 1.6000000000000001e168

   1. Initial program 93.6%

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

      1. sub-neg 93.6%

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

      2. neg-sub0 93.6%

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

      3. associate-+r- 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in eps around 0 80.3%

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

      1. neg-mul-1 80.3%

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

      2. rec-exp 80.3%

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

      3. *-commutative 80.3%

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

      4. neg-mul-1 80.3%

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

      5. rec-exp 80.3%

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

      6. distribute-lft1-in 80.3%

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

      7. rec-exp 80.3%

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

      8. distribute-lft-out 80.3%

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

      9. mul-1-neg 80.3%

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

      10. neg-mul-1 80.3%

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

      11. rec-exp 80.3%

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

      12. *-commutative 80.3%

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

      13. neg-mul-1 80.3%

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

      14. rec-exp 80.3%

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

      15. distribute-lft1-in 80.3%

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

      16. rec-exp 80.3%

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

   6. Simplified 80.3%

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

      1. exp-neg 80.3%

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

      2. un-div-inv 80.3%

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

   8. Applied egg-rr 80.3%

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

   if 1.32e46 < x < 6.5000000000000002e88 or 1.6000000000000001e168 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 40.7%

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

   5. Taylor expanded in eps around inf 40.9%

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

      1. exp-prod 40.9%

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

      2. *-commutative 40.9%

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

      3. sub-neg 40.9%

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

      4. neg-mul-1 40.9%

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

      5. *-commutative 40.9%

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

      6. exp-prod 40.9%

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

      7. +-commutative 40.9%

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

      8. associate-*r* 40.9%

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

      9. neg-mul-1 40.9%

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

      10. sub-neg 40.9%

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

      11. *-commutative 40.9%

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

      12. neg-mul-1 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 40:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+46} \lor \neg \left(x \leq 6.5 \cdot 10^{+88}\right) \land x \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 3: 58.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -0.5 \cdot \left(x \cdot x\right)\\ t_1 := \frac{2 + \varepsilon \cdot x}{2}\\ t_2 := \frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{if}\;x \leq -440:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;\frac{\frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77} \lor \neg \left(x \leq 5 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -0.5 (* x x))))
        (t_1 (/ (+ 2.0 (* eps x)) 2.0))
        (t_2
         (/
          (+
           2.0
           (*
            (- -1.0 (/ -1.0 eps))
            (/ x (/ (+ eps -1.0) (- 1.0 (* eps eps))))))
          2.0)))
   (if (<= x -440.0)
     (/ (/ (expm1 (- x)) eps) 2.0)
     (if (<= x 2.4e-182)
       (/ (+ t_0 t_0) 2.0)
       (if (<= x 1.2e-114)
         t_2
         (if (<= x 2.9e-62)
           t_1
           (if (<= x 1.75e-28)
             t_2
             (if (<= x 4.2)
               t_1
               (if (<= x 1e+43)
                 (/ (/ x (exp x)) 2.0)
                 (if (or (<= x 2e+77) (not (<= x 5e+169)))
                   (/ (/ (expm1 x) eps) 2.0)
                   0.0))))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (-0.5 * (x * x));
	double t_1 = (2.0 + (eps * x)) / 2.0;
	double t_2 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	double tmp;
	if (x <= -440.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 2.4e-182) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 1.2e-114) {
		tmp = t_2;
	} else if (x <= 2.9e-62) {
		tmp = t_1;
	} else if (x <= 1.75e-28) {
		tmp = t_2;
	} else if (x <= 4.2) {
		tmp = t_1;
	} else if (x <= 1e+43) {
		tmp = (x / exp(x)) / 2.0;
	} else if ((x <= 2e+77) || !(x <= 5e+169)) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (-0.5 * (x * x));
	double t_1 = (2.0 + (eps * x)) / 2.0;
	double t_2 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	double tmp;
	if (x <= -440.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 2.4e-182) {
		tmp = (t_0 + t_0) / 2.0;
	} else if (x <= 1.2e-114) {
		tmp = t_2;
	} else if (x <= 2.9e-62) {
		tmp = t_1;
	} else if (x <= 1.75e-28) {
		tmp = t_2;
	} else if (x <= 4.2) {
		tmp = t_1;
	} else if (x <= 1e+43) {
		tmp = (x / Math.exp(x)) / 2.0;
	} else if ((x <= 2e+77) || !(x <= 5e+169)) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + (-0.5 * (x * x))
	t_1 = (2.0 + (eps * x)) / 2.0
	t_2 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0
	tmp = 0
	if x <= -440.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 2.4e-182:
		tmp = (t_0 + t_0) / 2.0
	elif x <= 1.2e-114:
		tmp = t_2
	elif x <= 2.9e-62:
		tmp = t_1
	elif x <= 1.75e-28:
		tmp = t_2
	elif x <= 4.2:
		tmp = t_1
	elif x <= 1e+43:
		tmp = (x / math.exp(x)) / 2.0
	elif (x <= 2e+77) or not (x <= 5e+169):
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(-0.5 * Float64(x * x)))
	t_1 = Float64(Float64(2.0 + Float64(eps * x)) / 2.0)
	t_2 = Float64(Float64(2.0 + Float64(Float64(-1.0 - Float64(-1.0 / eps)) * Float64(x / Float64(Float64(eps + -1.0) / Float64(1.0 - Float64(eps * eps)))))) / 2.0)
	tmp = 0.0
	if (x <= -440.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 2.4e-182)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	elseif (x <= 1.2e-114)
		tmp = t_2;
	elseif (x <= 2.9e-62)
		tmp = t_1;
	elseif (x <= 1.75e-28)
		tmp = t_2;
	elseif (x <= 4.2)
		tmp = t_1;
	elseif (x <= 1e+43)
		tmp = Float64(Float64(x / exp(x)) / 2.0);
	elseif ((x <= 2e+77) || !(x <= 5e+169))
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(eps + -1.0), $MachinePrecision] / N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -440.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.4e-182], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.2e-114], t$95$2, If[LessEqual[x, 2.9e-62], t$95$1, If[LessEqual[x, 1.75e-28], t$95$2, If[LessEqual[x, 4.2], t$95$1, If[LessEqual[x, 1e+43], N[(N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2e+77], N[Not[LessEqual[x, 5e+169]], $MachinePrecision]], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -440

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 53.2%

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

   5. Taylor expanded in eps around 0 48.3%

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

      1. expm1-def 48.3%

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

      2. neg-mul-1 48.3%

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

   7. Simplified 48.3%

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

   if -440 < x < 2.3999999999999998e-182

   1. Initial program 47.8%

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

      1. sub-neg 47.8%

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

      2. neg-sub0 47.8%

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

      3. associate-+r- 47.8%

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

   3. Simplified 47.8%

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

   4. Taylor expanded in eps around 0 84.8%

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

      1. neg-mul-1 84.8%

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

      2. rec-exp 84.8%

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

      3. *-commutative 84.8%

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

      4. neg-mul-1 84.8%

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

      5. rec-exp 84.8%

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

      6. distribute-lft1-in 84.8%

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

      7. rec-exp 84.8%

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

      8. distribute-lft-out 84.8%

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

      9. mul-1-neg 84.8%

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

      10. neg-mul-1 84.8%

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

      11. rec-exp 84.8%

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

      12. *-commutative 84.8%

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

      13. neg-mul-1 84.8%

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

      14. rec-exp 84.8%

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

      15. distribute-lft1-in 84.8%

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

      16. rec-exp 84.8%

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

   6. Simplified 84.8%

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

   7. Taylor expanded in x around 0 84.6%

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

      1. unpow2 84.6%

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

   9. Simplified 84.6%

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

   10. Taylor expanded in x around 0 84.6%

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

       1. unpow2 84.6%

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

   12. Simplified 84.6%

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

   if 2.3999999999999998e-182 < x < 1.2000000000000001e-114 or 2.89999999999999986e-62 < x < 1.75e-28

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. neg-sub0 76.9%

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

      3. associate-+r- 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in x around 0 49.1%

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

   5. Taylor expanded in x around 0 26.0%

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

      1. *-commutative 26.0%

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

      2. flip-+ 57.7%

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

      3. associate-*r/ 57.7%

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

      4. metadata-eval 57.7%

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

      5. add-sqr-sqrt 48.4%

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

      6. sqrt-unprod 24.6%

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

      7. sqr-neg 24.6%

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

      8. sqrt-unprod 33.4%

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

      9. add-sqr-sqrt 47.8%

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

      10. neg-sub0 47.8%

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

      11. metadata-eval 47.8%

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

      12. associate--r- 47.8%

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

      13. metadata-eval 47.8%

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

      14. metadata-eval 47.8%

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

   7. Applied egg-rr 47.8%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   9. Simplified 47.8%

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

   if 1.2000000000000001e-114 < x < 2.89999999999999986e-62 or 1.75e-28 < x < 4.20000000000000018

   1. Initial program 56.4%

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

      1. sub-neg 56.4%

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

      2. neg-sub0 56.4%

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

      3. associate-+r- 56.4%

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

   3. Simplified 56.4%

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

   4. Taylor expanded in x around 0 32.2%

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

   5. Taylor expanded in eps around inf 75.8%

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

      1. exp-prod 75.8%

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

      2. *-commutative 75.8%

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

      3. sub-neg 75.8%

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

      4. neg-mul-1 75.8%

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

      5. *-commutative 75.8%

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

      6. exp-prod 75.8%

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

      7. +-commutative 75.8%

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

      8. associate-*r* 75.8%

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

      9. neg-mul-1 75.8%

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

      10. sub-neg 75.8%

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

      11. *-commutative 75.8%

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

      12. neg-mul-1 75.8%

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

   7. Simplified 75.8%

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

   8. Taylor expanded in eps around inf 75.8%

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

   9. Taylor expanded in eps around 0 65.5%

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

   if 4.20000000000000018 < x < 1.00000000000000001e43

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. neg-sub0 86.2%

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

      3. associate-+r- 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in eps around 0 71.9%

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

      1. neg-mul-1 71.9%

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

      2. rec-exp 71.9%

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

      3. *-commutative 71.9%

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

      4. neg-mul-1 71.9%

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

      5. rec-exp 71.9%

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

      6. distribute-lft1-in 71.9%

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

      7. rec-exp 71.9%

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

      8. distribute-lft-out 71.9%

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

      9. mul-1-neg 71.9%

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

      10. neg-mul-1 71.9%

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

      11. rec-exp 71.9%

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

      12. *-commutative 71.9%

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

      13. neg-mul-1 71.9%

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

      14. rec-exp 71.9%

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

      15. distribute-lft1-in 71.9%

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

      16. rec-exp 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in x around 0 3.9%

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

   8. Taylor expanded in x around inf 60.3%

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

      1. exp-neg 60.3%

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

      2. associate-*l/ 60.3%

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

      3. *-lft-identity 60.3%

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

   10. Simplified 60.3%

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

   if 1.00000000000000001e43 < x < 1.99999999999999997e77 or 5.00000000000000017e169 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 42.7%

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

   5. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. neg-mul-1 1.8%

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

   7. Simplified 1.8%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 41.2%

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

      5. sqr-neg 41.2%

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

      6. sqrt-unprod 41.2%

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

      7. add-sqr-sqrt 41.2%

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

   9. Applied egg-rr 41.2%

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

       1. expm1-def 41.2%

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

       2. expm1-log1p 41.5%

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

   11. Simplified 41.5%

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

   if 1.99999999999999997e77 < x < 5.00000000000000017e169

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 83.6%

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

      1. rec-exp 83.6%

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

      2. div-sub 83.6%

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

      3. rec-exp 83.6%

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

      4. neg-mul-1 83.6%

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

      5. rec-exp 83.6%

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

      6. neg-mul-1 83.6%

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

      7. +-inverses 83.6%

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

   6. Simplified 83.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 7 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -440:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right) + \left(1 + -0.5 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-62}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-28}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 4.2:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+43}:\\ \;\;\;\;\frac{\frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+77} \lor \neg \left(x \leq 5 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 52.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{if}\;x \leq 3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-62}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{+46}:\\ \;\;\;\;\frac{\frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+77} \lor \neg \left(x \leq 1.65 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/
          (+
           2.0
           (*
            (- -1.0 (/ -1.0 eps))
            (/ x (/ (+ eps -1.0) (- 1.0 (* eps eps))))))
          2.0)))
   (if (<= x 3.6e-185)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x 2.8e-114)
       t_0
       (if (<= x 1.32e-62)
         (/ (+ 2.0 (* eps x)) 2.0)
         (if (<= x 9.5e-25)
           t_0
           (if (<= x 1e+46)
             (/ (/ x (exp x)) 2.0)
             (if (or (<= x 3e+77) (not (<= x 1.65e+168)))
               (/ (/ (expm1 x) eps) 2.0)
               0.0))))))))

C

double code(double x, double eps) {
	double t_0 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	double tmp;
	if (x <= 3.6e-185) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.8e-114) {
		tmp = t_0;
	} else if (x <= 1.32e-62) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 9.5e-25) {
		tmp = t_0;
	} else if (x <= 1e+46) {
		tmp = (x / exp(x)) / 2.0;
	} else if ((x <= 3e+77) || !(x <= 1.65e+168)) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	double tmp;
	if (x <= 3.6e-185) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.8e-114) {
		tmp = t_0;
	} else if (x <= 1.32e-62) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 9.5e-25) {
		tmp = t_0;
	} else if (x <= 1e+46) {
		tmp = (x / Math.exp(x)) / 2.0;
	} else if ((x <= 3e+77) || !(x <= 1.65e+168)) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0
	tmp = 0
	if x <= 3.6e-185:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 2.8e-114:
		tmp = t_0
	elif x <= 1.32e-62:
		tmp = (2.0 + (eps * x)) / 2.0
	elif x <= 9.5e-25:
		tmp = t_0
	elif x <= 1e+46:
		tmp = (x / math.exp(x)) / 2.0
	elif (x <= 3e+77) or not (x <= 1.65e+168):
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(2.0 + Float64(Float64(-1.0 - Float64(-1.0 / eps)) * Float64(x / Float64(Float64(eps + -1.0) / Float64(1.0 - Float64(eps * eps)))))) / 2.0)
	tmp = 0.0
	if (x <= 3.6e-185)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 2.8e-114)
		tmp = t_0;
	elseif (x <= 1.32e-62)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	elseif (x <= 9.5e-25)
		tmp = t_0;
	elseif (x <= 1e+46)
		tmp = Float64(Float64(x / exp(x)) / 2.0);
	elseif ((x <= 3e+77) || !(x <= 1.65e+168))
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 + N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(eps + -1.0), $MachinePrecision] / N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 3.6e-185], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e-114], t$95$0, If[LessEqual[x, 1.32e-62], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.5e-25], t$95$0, If[LessEqual[x, 1e+46], N[(N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3e+77], N[Not[LessEqual[x, 1.65e+168]], $MachinePrecision]], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 3.5999999999999998e-185

   1. Initial program 58.3%

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

      1. sub-neg 58.3%

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

      2. neg-sub0 58.3%

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

      3. associate-+r- 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around 0 43.0%

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

   5. Taylor expanded in x around 0 49.3%

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

   6. Taylor expanded in eps around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. distribute-lft-neg-out 73.4%

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

      3. *-commutative 73.4%

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

   8. Simplified 73.4%

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

   if 3.5999999999999998e-185 < x < 2.8000000000000001e-114 or 1.31999999999999997e-62 < x < 9.50000000000000065e-25

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. neg-sub0 76.9%

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

      3. associate-+r- 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in x around 0 49.1%

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

   5. Taylor expanded in x around 0 26.0%

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

      1. *-commutative 26.0%

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

      2. flip-+ 57.7%

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

      3. associate-*r/ 57.7%

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

      4. metadata-eval 57.7%

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

      5. add-sqr-sqrt 48.4%

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

      6. sqrt-unprod 24.6%

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

      7. sqr-neg 24.6%

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

      8. sqrt-unprod 33.4%

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

      9. add-sqr-sqrt 47.8%

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

      10. neg-sub0 47.8%

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

      11. metadata-eval 47.8%

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

      12. associate--r- 47.8%

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

      13. metadata-eval 47.8%

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

      14. metadata-eval 47.8%

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

   7. Applied egg-rr 47.8%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   9. Simplified 47.8%

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

   if 2.8000000000000001e-114 < x < 1.31999999999999997e-62

   1. Initial program 54.4%

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

      1. sub-neg 54.4%

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

      2. neg-sub0 54.4%

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

      3. associate-+r- 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in x around 0 31.6%

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

   5. Taylor expanded in eps around inf 77.2%

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

      1. exp-prod 77.2%

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

      2. *-commutative 77.2%

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

      3. sub-neg 77.2%

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

      4. neg-mul-1 77.2%

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

      5. *-commutative 77.2%

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

      6. exp-prod 77.2%

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

      7. +-commutative 77.2%

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

      8. associate-*r* 77.2%

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

      9. neg-mul-1 77.2%

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

      10. sub-neg 77.2%

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

      11. *-commutative 77.2%

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

      12. neg-mul-1 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in eps around inf 77.2%

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

   9. Taylor expanded in eps around 0 70.9%

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

   if 9.50000000000000065e-25 < x < 9.9999999999999999e45

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. neg-sub0 82.9%

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

      3. associate-+r- 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in eps around 0 65.4%

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

      1. neg-mul-1 65.4%

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

      2. rec-exp 65.4%

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

      3. *-commutative 65.4%

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

      4. neg-mul-1 65.4%

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

      5. rec-exp 65.4%

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

      6. distribute-lft1-in 65.4%

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

      7. rec-exp 65.4%

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

      8. distribute-lft-out 65.4%

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

      9. mul-1-neg 65.4%

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

      10. neg-mul-1 65.4%

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

      11. rec-exp 65.4%

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

      12. *-commutative 65.4%

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

      13. neg-mul-1 65.4%

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

      14. rec-exp 65.4%

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

      15. distribute-lft1-in 65.4%

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

      16. rec-exp 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in x around 0 9.4%

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

   8. Taylor expanded in x around inf 50.5%

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

      1. exp-neg 50.5%

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

      2. associate-*l/ 50.5%

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

      3. *-lft-identity 50.5%

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

   10. Simplified 50.5%

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

   if 9.9999999999999999e45 < x < 2.9999999999999998e77 or 1.6499999999999999e168 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 42.7%

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

   5. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. neg-mul-1 1.8%

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

   7. Simplified 1.8%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 41.2%

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

      5. sqr-neg 41.2%

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

      6. sqrt-unprod 41.2%

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

      7. add-sqr-sqrt 41.2%

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

   9. Applied egg-rr 41.2%

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

       1. expm1-def 41.2%

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

       2. expm1-log1p 41.5%

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

   11. Simplified 41.5%

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

   if 2.9999999999999998e77 < x < 1.6499999999999999e168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 83.6%

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

      1. rec-exp 83.6%

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

      2. div-sub 83.6%

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

      3. rec-exp 83.6%

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

      4. neg-mul-1 83.6%

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

      5. rec-exp 83.6%

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

      6. neg-mul-1 83.6%

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

      7. +-inverses 83.6%

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

   6. Simplified 83.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-62}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 10^{+46}:\\ \;\;\;\;\frac{\frac{x}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+77} \lor \neg \left(x \leq 1.65 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 65.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x - \varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14} \lor \neg \left(x \leq 3 \cdot 10^{+46} \lor \neg \left(x \leq 1.4 \cdot 10^{+90}\right) \land x \leq 10^{+168}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-299)
   (/ (+ 1.0 (exp (- x (* eps x)))) 2.0)
   (if (or (<= x 5e+14)
           (not (or (<= x 3e+46) (and (not (<= x 1.4e+90)) (<= x 1e+168)))))
     (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-299) {
		tmp = (1.0 + exp((x - (eps * x)))) / 2.0;
	} else if ((x <= 5e+14) || !((x <= 3e+46) || (!(x <= 1.4e+90) && (x <= 1e+168)))) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-4d-299)) then
        tmp = (1.0d0 + exp((x - (eps * x)))) / 2.0d0
    else if ((x <= 5d+14) .or. (.not. (x <= 3d+46) .or. (.not. (x <= 1.4d+90)) .and. (x <= 1d+168))) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 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 <= -4e-299) {
		tmp = (1.0 + Math.exp((x - (eps * x)))) / 2.0;
	} else if ((x <= 5e+14) || !((x <= 3e+46) || (!(x <= 1.4e+90) && (x <= 1e+168)))) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4e-299:
		tmp = (1.0 + math.exp((x - (eps * x)))) / 2.0
	elif (x <= 5e+14) or not ((x <= 3e+46) or (not (x <= 1.4e+90) and (x <= 1e+168))):
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-299)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(eps * x)))) / 2.0);
	elseif ((x <= 5e+14) || !((x <= 3e+46) || (!(x <= 1.4e+90) && (x <= 1e+168))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4e-299)
		tmp = (1.0 + exp((x - (eps * x)))) / 2.0;
	elseif ((x <= 5e+14) || ~(((x <= 3e+46) || (~((x <= 1.4e+90)) && (x <= 1e+168)))))
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-299], N[(N[(1.0 + N[Exp[N[(x - N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5e+14], N[Not[Or[LessEqual[x, 3e+46], And[N[Not[LessEqual[x, 1.4e+90]], $MachinePrecision], LessEqual[x, 1e+168]]]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999997e-299

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. neg-sub0 61.8%

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

      3. associate-+r- 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in eps around inf 76.2%

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

      1. exp-prod 76.2%

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

      2. *-commutative 76.2%

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

      3. sub-neg 76.2%

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

      4. neg-mul-1 76.2%

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

      5. *-commutative 76.2%

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

      6. exp-prod 76.2%

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

      7. +-commutative 76.2%

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

      8. associate-*r* 76.2%

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

      9. neg-mul-1 76.2%

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

      10. sub-neg 76.2%

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

      11. *-commutative 76.2%

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

      12. neg-mul-1 76.2%

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

   7. Simplified 76.2%

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

      1. add-sqr-sqrt 9.0%

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

      2. sqrt-unprod 54.9%

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

      3. sqr-neg 54.9%

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

      4. sqrt-unprod 48.5%

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

      5. add-sqr-sqrt 79.7%

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

      6. sub-neg 79.7%

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

      7. distribute-lft-in 79.7%

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

      8. *-rgt-identity 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. distribute-rgt-neg-out 79.7%

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

       2. unsub-neg 79.7%

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

       3. *-commutative 79.7%

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

   11. Simplified 79.7%

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

   if -3.99999999999999997e-299 < x < 5e14 or 3.00000000000000023e46 < x < 1.4e90 or 9.9999999999999993e167 < x

   1. Initial program 72.2%

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

      1. sub-neg 72.2%

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

      2. neg-sub0 72.2%

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

      3. associate-+r- 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x around 0 41.1%

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

   5. Taylor expanded in eps around inf 67.5%

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

      1. exp-prod 67.5%

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

      2. *-commutative 67.5%

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

      3. sub-neg 67.5%

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

      4. neg-mul-1 67.5%

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

      5. *-commutative 67.5%

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

      6. exp-prod 67.5%

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

      7. +-commutative 67.5%

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

      8. associate-*r* 67.5%

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

      9. neg-mul-1 67.5%

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

      10. sub-neg 67.5%

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

      11. *-commutative 67.5%

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

      12. neg-mul-1 67.5%

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

   7. Simplified 67.5%

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

   if 5e14 < x < 3.00000000000000023e46 or 1.4e90 < x < 9.9999999999999993e167

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 84.3%

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

      1. rec-exp 84.3%

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

      2. div-sub 84.3%

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

      3. rec-exp 84.3%

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

      4. neg-mul-1 84.3%

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

      5. rec-exp 84.3%

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

      6. neg-mul-1 84.3%

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

      7. +-inverses 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x - \varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+14} \lor \neg \left(x \leq 3 \cdot 10^{+46} \lor \neg \left(x \leq 1.4 \cdot 10^{+90}\right) \land x \leq 10^{+168}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 65.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 48000000000000 \lor \neg \left(x \leq 1.02 \cdot 10^{+43}\right) \land \left(x \leq 2.7 \cdot 10^{+90} \lor \neg \left(x \leq 1.05 \cdot 10^{+168}\right)\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-299)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (or (<= x 48000000000000.0)
           (and (not (<= x 1.02e+43))
                (or (<= x 2.7e+90) (not (<= x 1.05e+168)))))
     (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-299) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 48000000000000.0) || (!(x <= 1.02e+43) && ((x <= 2.7e+90) || !(x <= 1.05e+168)))) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-4d-299)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 48000000000000.0d0) .or. (.not. (x <= 1.02d+43)) .and. (x <= 2.7d+90) .or. (.not. (x <= 1.05d+168))) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 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 <= -4e-299) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 48000000000000.0) || (!(x <= 1.02e+43) && ((x <= 2.7e+90) || !(x <= 1.05e+168)))) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4e-299:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 48000000000000.0) or (not (x <= 1.02e+43) and ((x <= 2.7e+90) or not (x <= 1.05e+168))):
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-299)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 48000000000000.0) || (!(x <= 1.02e+43) && ((x <= 2.7e+90) || !(x <= 1.05e+168))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4e-299)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 48000000000000.0) || (~((x <= 1.02e+43)) && ((x <= 2.7e+90) || ~((x <= 1.05e+168)))))
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-299], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 48000000000000.0], And[N[Not[LessEqual[x, 1.02e+43]], $MachinePrecision], Or[LessEqual[x, 2.7e+90], N[Not[LessEqual[x, 1.05e+168]], $MachinePrecision]]]], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999997e-299

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. neg-sub0 61.8%

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

      3. associate-+r- 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 42.4%

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

   5. Taylor expanded in eps around inf 79.7%

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

      1. mul-1-neg 79.7%

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

      2. exp-prod 79.7%

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

      3. +-commutative 79.7%

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

      4. remove-double-neg 79.7%

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

      5. sub-neg 79.7%

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

      6. neg-mul-1 79.7%

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

      7. exp-prod 79.7%

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

      8. mul-1-neg 79.7%

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

      9. *-commutative 79.7%

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

      10. sub-neg 79.7%

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

      11. neg-mul-1 79.7%

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

      12. remove-double-neg 79.7%

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

   7. Simplified 79.7%

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

   if -3.99999999999999997e-299 < x < 4.8e13 or 1.02e43 < x < 2.7e90 or 1.05000000000000001e168 < x

   1. Initial program 72.2%

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

      1. sub-neg 72.2%

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

      2. neg-sub0 72.2%

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

      3. associate-+r- 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x around 0 41.1%

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

   5. Taylor expanded in eps around inf 67.5%

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

      1. exp-prod 67.5%

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

      2. *-commutative 67.5%

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

      3. sub-neg 67.5%

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

      4. neg-mul-1 67.5%

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

      5. *-commutative 67.5%

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

      6. exp-prod 67.5%

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

      7. +-commutative 67.5%

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

      8. associate-*r* 67.5%

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

      9. neg-mul-1 67.5%

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

      10. sub-neg 67.5%

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

      11. *-commutative 67.5%

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

      12. neg-mul-1 67.5%

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

   7. Simplified 67.5%

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

   if 4.8e13 < x < 1.02e43 or 2.7e90 < x < 1.05000000000000001e168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 84.3%

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

      1. rec-exp 84.3%

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

      2. div-sub 84.3%

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

      3. rec-exp 84.3%

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

      4. neg-mul-1 84.3%

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

      5. rec-exp 84.3%

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

      6. neg-mul-1 84.3%

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

      7. +-inverses 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 48000000000000 \lor \neg \left(x \leq 1.02 \cdot 10^{+43}\right) \land \left(x \leq 2.7 \cdot 10^{+90} \lor \neg \left(x \leq 1.05 \cdot 10^{+168}\right)\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 350000000000 \lor \neg \left(x \leq 1.02 \cdot 10^{+43}\right) \land \left(x \leq 1.1 \cdot 10^{+91} \lor \neg \left(x \leq 10^{+168}\right)\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -350.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (or (<= x 350000000000.0)
           (and (not (<= x 1.02e+43)) (or (<= x 1.1e+91) (not (<= x 1e+168)))))
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -350.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if ((x <= 350000000000.0) || (!(x <= 1.02e+43) && ((x <= 1.1e+91) || !(x <= 1e+168)))) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -350.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if ((x <= 350000000000.0) || (!(x <= 1.02e+43) && ((x <= 1.1e+91) || !(x <= 1e+168)))) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -350.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif (x <= 350000000000.0) or (not (x <= 1.02e+43) and ((x <= 1.1e+91) or not (x <= 1e+168))):
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -350.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif ((x <= 350000000000.0) || (!(x <= 1.02e+43) && ((x <= 1.1e+91) || !(x <= 1e+168))))
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -350.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 350000000000.0], And[N[Not[LessEqual[x, 1.02e+43]], $MachinePrecision], Or[LessEqual[x, 1.1e+91], N[Not[LessEqual[x, 1e+168]], $MachinePrecision]]]], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -350

   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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 53.2%

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

   5. Taylor expanded in eps around 0 48.3%

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

      1. expm1-def 48.3%

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

      2. neg-mul-1 48.3%

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

   7. Simplified 48.3%

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

   if -350 < x < 3.5e11 or 1.02e43 < x < 1.1e91 or 9.9999999999999993e167 < x

   1. Initial program 62.6%

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

      1. sub-neg 62.6%

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

      2. neg-sub0 62.6%

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

      3. associate-+r- 62.6%

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

   3. Simplified 62.6%

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

   4. Taylor expanded in x around 0 38.2%

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

   5. Taylor expanded in eps around inf 74.2%

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

      1. exp-prod 74.2%

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

      2. *-commutative 74.2%

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

      3. sub-neg 74.2%

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

      4. neg-mul-1 74.2%

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

      5. *-commutative 74.2%

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

      6. exp-prod 74.2%

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

      7. +-commutative 74.2%

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

      8. associate-*r* 74.2%

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

      9. neg-mul-1 74.2%

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

      10. sub-neg 74.2%

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

      11. *-commutative 74.2%

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

      12. neg-mul-1 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in eps around inf 74.5%

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

   if 3.5e11 < x < 1.02e43 or 1.1e91 < x < 9.9999999999999993e167

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 84.3%

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

      1. rec-exp 84.3%

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

      2. div-sub 84.3%

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

      3. rec-exp 84.3%

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

      4. neg-mul-1 84.3%

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

      5. rec-exp 84.3%

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

      6. neg-mul-1 84.3%

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

      7. +-inverses 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 350000000000 \lor \neg \left(x \leq 1.02 \cdot 10^{+43}\right) \land \left(x \leq 1.1 \cdot 10^{+91} \lor \neg \left(x \leq 10^{+168}\right)\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x - \varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+14} \lor \neg \left(x \leq 6 \cdot 10^{+45}\right) \land \left(x \leq 1.05 \cdot 10^{+91} \lor \neg \left(x \leq 2.9 \cdot 10^{+168}\right)\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-299)
   (/ (+ 1.0 (exp (- x (* eps x)))) 2.0)
   (if (or (<= x 2.6e+14)
           (and (not (<= x 6e+45)) (or (<= x 1.05e+91) (not (<= x 2.9e+168)))))
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-299) {
		tmp = (1.0 + exp((x - (eps * x)))) / 2.0;
	} else if ((x <= 2.6e+14) || (!(x <= 6e+45) && ((x <= 1.05e+91) || !(x <= 2.9e+168)))) {
		tmp = (1.0 + exp((eps * 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 <= (-4d-299)) then
        tmp = (1.0d0 + exp((x - (eps * x)))) / 2.0d0
    else if ((x <= 2.6d+14) .or. (.not. (x <= 6d+45)) .and. (x <= 1.05d+91) .or. (.not. (x <= 2.9d+168))) then
        tmp = (1.0d0 + exp((eps * 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 <= -4e-299) {
		tmp = (1.0 + Math.exp((x - (eps * x)))) / 2.0;
	} else if ((x <= 2.6e+14) || (!(x <= 6e+45) && ((x <= 1.05e+91) || !(x <= 2.9e+168)))) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4e-299:
		tmp = (1.0 + math.exp((x - (eps * x)))) / 2.0
	elif (x <= 2.6e+14) or (not (x <= 6e+45) and ((x <= 1.05e+91) or not (x <= 2.9e+168))):
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-299)
		tmp = Float64(Float64(1.0 + exp(Float64(x - Float64(eps * x)))) / 2.0);
	elseif ((x <= 2.6e+14) || (!(x <= 6e+45) && ((x <= 1.05e+91) || !(x <= 2.9e+168))))
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4e-299)
		tmp = (1.0 + exp((x - (eps * x)))) / 2.0;
	elseif ((x <= 2.6e+14) || (~((x <= 6e+45)) && ((x <= 1.05e+91) || ~((x <= 2.9e+168)))))
		tmp = (1.0 + exp((eps * x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-299], N[(N[(1.0 + N[Exp[N[(x - N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.6e+14], And[N[Not[LessEqual[x, 6e+45]], $MachinePrecision], Or[LessEqual[x, 1.05e+91], N[Not[LessEqual[x, 2.9e+168]], $MachinePrecision]]]], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999997e-299

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. neg-sub0 61.8%

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

      3. associate-+r- 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in eps around inf 76.2%

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

      1. exp-prod 76.2%

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

      2. *-commutative 76.2%

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

      3. sub-neg 76.2%

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

      4. neg-mul-1 76.2%

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

      5. *-commutative 76.2%

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

      6. exp-prod 76.2%

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

      7. +-commutative 76.2%

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

      8. associate-*r* 76.2%

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

      9. neg-mul-1 76.2%

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

      10. sub-neg 76.2%

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

      11. *-commutative 76.2%

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

      12. neg-mul-1 76.2%

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

   7. Simplified 76.2%

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

      1. add-sqr-sqrt 9.0%

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

      2. sqrt-unprod 54.9%

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

      3. sqr-neg 54.9%

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

      4. sqrt-unprod 48.5%

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

      5. add-sqr-sqrt 79.7%

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

      6. sub-neg 79.7%

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

      7. distribute-lft-in 79.7%

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

      8. *-rgt-identity 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. distribute-rgt-neg-out 79.7%

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

       2. unsub-neg 79.7%

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

       3. *-commutative 79.7%

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

   11. Simplified 79.7%

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

   if -3.99999999999999997e-299 < x < 2.6e14 or 6.00000000000000021e45 < x < 1.05000000000000004e91 or 2.9e168 < x

   1. Initial program 72.2%

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

      1. sub-neg 72.2%

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

      2. neg-sub0 72.2%

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

      3. associate-+r- 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x around 0 41.1%

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

   5. Taylor expanded in eps around inf 67.5%

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

      1. exp-prod 67.5%

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

      2. *-commutative 67.5%

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

      3. sub-neg 67.5%

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

      4. neg-mul-1 67.5%

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

      5. *-commutative 67.5%

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

      6. exp-prod 67.5%

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

      7. +-commutative 67.5%

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

      8. associate-*r* 67.5%

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

      9. neg-mul-1 67.5%

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

      10. sub-neg 67.5%

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

      11. *-commutative 67.5%

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

      12. neg-mul-1 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in eps around inf 67.5%

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

   if 2.6e14 < x < 6.00000000000000021e45 or 1.05000000000000004e91 < x < 2.9e168

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 84.3%

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

      1. rec-exp 84.3%

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

      2. div-sub 84.3%

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

      3. rec-exp 84.3%

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

      4. neg-mul-1 84.3%

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

      5. rec-exp 84.3%

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

      6. neg-mul-1 84.3%

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

      7. +-inverses 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-299}:\\ \;\;\;\;\frac{1 + e^{x - \varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+14} \lor \neg \left(x \leq 6 \cdot 10^{+45}\right) \land \left(x \leq 1.05 \cdot 10^{+91} \lor \neg \left(x \leq 2.9 \cdot 10^{+168}\right)\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 53.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{-1}{\varepsilon}\\ t_1 := 1 - \varepsilon \cdot \varepsilon\\ t_2 := \frac{2 + t_0 \cdot \frac{x}{\frac{\varepsilon + -1}{t_1}}}{2}\\ \mathbf{if}\;x \leq 1.06 \cdot 10^{-184}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-111}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{x}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \frac{x \cdot t_1}{\varepsilon + -1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ -1.0 eps)))
        (t_1 (- 1.0 (* eps eps)))
        (t_2 (/ (+ 2.0 (* t_0 (/ x (/ (+ eps -1.0) t_1)))) 2.0)))
   (if (<= x 1.06e-184)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x 5.4e-111)
       t_2
       (if (<= x 7.2e-59)
         (/ (+ 2.0 (* eps x)) 2.0)
         (if (<= x 9.5e-25)
           t_2
           (if (<= x 8.5e+169)
             (/ (/ x (exp x)) 2.0)
             (/ (+ 2.0 (* t_0 (/ (* x t_1) (+ eps -1.0)))) 2.0))))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 - (-1.0 / eps);
	double t_1 = 1.0 - (eps * eps);
	double t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0;
	double tmp;
	if (x <= 1.06e-184) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 5.4e-111) {
		tmp = t_2;
	} else if (x <= 7.2e-59) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 9.5e-25) {
		tmp = t_2;
	} else if (x <= 8.5e+169) {
		tmp = (x / exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) - ((-1.0d0) / eps)
    t_1 = 1.0d0 - (eps * eps)
    t_2 = (2.0d0 + (t_0 * (x / ((eps + (-1.0d0)) / t_1)))) / 2.0d0
    if (x <= 1.06d-184) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 5.4d-111) then
        tmp = t_2
    else if (x <= 7.2d-59) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else if (x <= 9.5d-25) then
        tmp = t_2
    else if (x <= 8.5d+169) then
        tmp = (x / exp(x)) / 2.0d0
    else
        tmp = (2.0d0 + (t_0 * ((x * t_1) / (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 - (-1.0 / eps);
	double t_1 = 1.0 - (eps * eps);
	double t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0;
	double tmp;
	if (x <= 1.06e-184) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 5.4e-111) {
		tmp = t_2;
	} else if (x <= 7.2e-59) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 9.5e-25) {
		tmp = t_2;
	} else if (x <= 8.5e+169) {
		tmp = (x / Math.exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 - (-1.0 / eps)
	t_1 = 1.0 - (eps * eps)
	t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0
	tmp = 0
	if x <= 1.06e-184:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 5.4e-111:
		tmp = t_2
	elif x <= 7.2e-59:
		tmp = (2.0 + (eps * x)) / 2.0
	elif x <= 9.5e-25:
		tmp = t_2
	elif x <= 8.5e+169:
		tmp = (x / math.exp(x)) / 2.0
	else:
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 - Float64(-1.0 / eps))
	t_1 = Float64(1.0 - Float64(eps * eps))
	t_2 = Float64(Float64(2.0 + Float64(t_0 * Float64(x / Float64(Float64(eps + -1.0) / t_1)))) / 2.0)
	tmp = 0.0
	if (x <= 1.06e-184)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 5.4e-111)
		tmp = t_2;
	elseif (x <= 7.2e-59)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	elseif (x <= 9.5e-25)
		tmp = t_2;
	elseif (x <= 8.5e+169)
		tmp = Float64(Float64(x / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(x * t_1) / Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 - (-1.0 / eps);
	t_1 = 1.0 - (eps * eps);
	t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0;
	tmp = 0.0;
	if (x <= 1.06e-184)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 5.4e-111)
		tmp = t_2;
	elseif (x <= 7.2e-59)
		tmp = (2.0 + (eps * x)) / 2.0;
	elseif (x <= 9.5e-25)
		tmp = t_2;
	elseif (x <= 8.5e+169)
		tmp = (x / exp(x)) / 2.0;
	else
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(t$95$0 * N[(x / N[(N[(eps + -1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 1.06e-184], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.4e-111], t$95$2, If[LessEqual[x, 7.2e-59], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.5e-25], t$95$2, If[LessEqual[x, 8.5e+169], N[(N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(t$95$0 * N[(N[(x * t$95$1), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.05999999999999995e-184

   1. Initial program 58.3%

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

      1. sub-neg 58.3%

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

      2. neg-sub0 58.3%

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

      3. associate-+r- 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around 0 43.0%

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

   5. Taylor expanded in x around 0 49.3%

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

   6. Taylor expanded in eps around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. distribute-lft-neg-out 73.4%

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

      3. *-commutative 73.4%

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

   8. Simplified 73.4%

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

   if 1.05999999999999995e-184 < x < 5.39999999999999977e-111 or 7.20000000000000001e-59 < x < 9.50000000000000065e-25

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. neg-sub0 76.9%

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

      3. associate-+r- 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in x around 0 49.1%

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

   5. Taylor expanded in x around 0 26.0%

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

      1. *-commutative 26.0%

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

      2. flip-+ 57.7%

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

      3. associate-*r/ 57.7%

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

      4. metadata-eval 57.7%

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

      5. add-sqr-sqrt 48.4%

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

      6. sqrt-unprod 24.6%

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

      7. sqr-neg 24.6%

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

      8. sqrt-unprod 33.4%

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

      9. add-sqr-sqrt 47.8%

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

      10. neg-sub0 47.8%

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

      11. metadata-eval 47.8%

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

      12. associate--r- 47.8%

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

      13. metadata-eval 47.8%

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

      14. metadata-eval 47.8%

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

   7. Applied egg-rr 47.8%

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

      1. associate-/l* 47.8%

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

      2. +-commutative 47.8%

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

   9. Simplified 47.8%

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

   if 5.39999999999999977e-111 < x < 7.20000000000000001e-59

   1. Initial program 54.4%

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

      1. sub-neg 54.4%

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

      2. neg-sub0 54.4%

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

      3. associate-+r- 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in x around 0 31.6%

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

   5. Taylor expanded in eps around inf 77.2%

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

      1. exp-prod 77.2%

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

      2. *-commutative 77.2%

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

      3. sub-neg 77.2%

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

      4. neg-mul-1 77.2%

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

      5. *-commutative 77.2%

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

      6. exp-prod 77.2%

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

      7. +-commutative 77.2%

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

      8. associate-*r* 77.2%

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

      9. neg-mul-1 77.2%

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

      10. sub-neg 77.2%

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

      11. *-commutative 77.2%

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

      12. neg-mul-1 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in eps around inf 77.2%

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

   9. Taylor expanded in eps around 0 70.9%

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

   if 9.50000000000000065e-25 < x < 8.5000000000000004e169

   1. Initial program 93.8%

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

      1. sub-neg 93.8%

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

      2. neg-sub0 93.8%

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

      3. associate-+r- 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in eps around 0 62.4%

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

      1. neg-mul-1 62.4%

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

      2. rec-exp 62.4%

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

      3. *-commutative 62.4%

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

      4. neg-mul-1 62.4%

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

      5. rec-exp 62.4%

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

      6. distribute-lft1-in 62.4%

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

      7. rec-exp 62.4%

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

      8. distribute-lft-out 62.4%

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

      9. mul-1-neg 62.4%

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

      10. neg-mul-1 62.4%

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

      11. rec-exp 62.4%

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

      12. *-commutative 62.4%

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

      13. neg-mul-1 62.4%

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

      14. rec-exp 62.4%

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

      15. distribute-lft1-in 62.4%

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

      16. rec-exp 62.4%

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

   6. Simplified 62.4%

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

   7. Taylor expanded in x around 0 5.4%

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

   8. Taylor expanded in x around inf 57.0%

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

      1. exp-neg 57.0%

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

      2. associate-*l/ 57.0%

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

      3. *-lft-identity 57.0%

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

   10. Simplified 57.0%

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

   if 8.5000000000000004e169 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 24.4%

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

   5. Taylor expanded in x around 0 19.4%

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

      1. flip-+ 19.4%

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

      2. associate-*l/ 22.8%

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

      3. metadata-eval 22.8%

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

      4. add-sqr-sqrt 22.8%

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

      5. sqrt-unprod 31.0%

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

      6. sqr-neg 31.0%

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

      7. sqrt-unprod 34.2%

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

      8. add-sqr-sqrt 34.7%

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

      9. neg-sub0 34.7%

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

      10. metadata-eval 34.7%

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

      11. associate--r- 34.7%

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

      12. metadata-eval 34.7%

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

      13. metadata-eval 34.7%

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

   7. Applied egg-rr 34.7%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.06 \cdot 10^{-184}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-111}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{\frac{x}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\varepsilon + -1}}{2}\\ \end{array} \]

Alternative 10: 53.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{-1}{\varepsilon}\\ t_1 := 1 - \varepsilon \cdot \varepsilon\\ t_2 := \frac{2 + t_0 \cdot \frac{x}{\frac{\varepsilon + -1}{t_1}}}{2}\\ \mathbf{if}\;x \leq 1.25 \cdot 10^{-183}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-114}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-66}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 45:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + t_0 \cdot \frac{x \cdot t_1}{\varepsilon + -1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ -1.0 eps)))
        (t_1 (- 1.0 (* eps eps)))
        (t_2 (/ (+ 2.0 (* t_0 (/ x (/ (+ eps -1.0) t_1)))) 2.0)))
   (if (<= x 1.25e-183)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x 1.35e-114)
       t_2
       (if (<= x 6e-66)
         (/ (+ 2.0 (* eps x)) 2.0)
         (if (<= x 45.0)
           t_2
           (if (<= x 3.9e+170)
             0.0
             (/ (+ 2.0 (* t_0 (/ (* x t_1) (+ eps -1.0)))) 2.0))))))))

C

double code(double x, double eps) {
	double t_0 = -1.0 - (-1.0 / eps);
	double t_1 = 1.0 - (eps * eps);
	double t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0;
	double tmp;
	if (x <= 1.25e-183) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 1.35e-114) {
		tmp = t_2;
	} else if (x <= 6e-66) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 45.0) {
		tmp = t_2;
	} else if (x <= 3.9e+170) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-1.0d0) - ((-1.0d0) / eps)
    t_1 = 1.0d0 - (eps * eps)
    t_2 = (2.0d0 + (t_0 * (x / ((eps + (-1.0d0)) / t_1)))) / 2.0d0
    if (x <= 1.25d-183) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 1.35d-114) then
        tmp = t_2
    else if (x <= 6d-66) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else if (x <= 45.0d0) then
        tmp = t_2
    else if (x <= 3.9d+170) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (t_0 * ((x * t_1) / (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 - (-1.0 / eps);
	double t_1 = 1.0 - (eps * eps);
	double t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0;
	double tmp;
	if (x <= 1.25e-183) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 1.35e-114) {
		tmp = t_2;
	} else if (x <= 6e-66) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 45.0) {
		tmp = t_2;
	} else if (x <= 3.9e+170) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 - (-1.0 / eps)
	t_1 = 1.0 - (eps * eps)
	t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0
	tmp = 0
	if x <= 1.25e-183:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 1.35e-114:
		tmp = t_2
	elif x <= 6e-66:
		tmp = (2.0 + (eps * x)) / 2.0
	elif x <= 45.0:
		tmp = t_2
	elif x <= 3.9e+170:
		tmp = 0.0
	else:
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 - Float64(-1.0 / eps))
	t_1 = Float64(1.0 - Float64(eps * eps))
	t_2 = Float64(Float64(2.0 + Float64(t_0 * Float64(x / Float64(Float64(eps + -1.0) / t_1)))) / 2.0)
	tmp = 0.0
	if (x <= 1.25e-183)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 1.35e-114)
		tmp = t_2;
	elseif (x <= 6e-66)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	elseif (x <= 45.0)
		tmp = t_2;
	elseif (x <= 3.9e+170)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(t_0 * Float64(Float64(x * t_1) / Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 - (-1.0 / eps);
	t_1 = 1.0 - (eps * eps);
	t_2 = (2.0 + (t_0 * (x / ((eps + -1.0) / t_1)))) / 2.0;
	tmp = 0.0;
	if (x <= 1.25e-183)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 1.35e-114)
		tmp = t_2;
	elseif (x <= 6e-66)
		tmp = (2.0 + (eps * x)) / 2.0;
	elseif (x <= 45.0)
		tmp = t_2;
	elseif (x <= 3.9e+170)
		tmp = 0.0;
	else
		tmp = (2.0 + (t_0 * ((x * t_1) / (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(t$95$0 * N[(x / N[(N[(eps + -1.0), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 1.25e-183], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.35e-114], t$95$2, If[LessEqual[x, 6e-66], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 45.0], t$95$2, If[LessEqual[x, 3.9e+170], 0.0, N[(N[(2.0 + N[(t$95$0 * N[(N[(x * t$95$1), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.2500000000000001e-183

   1. Initial program 58.3%

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

      1. sub-neg 58.3%

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

      2. neg-sub0 58.3%

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

      3. associate-+r- 58.3%

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

   3. Simplified 58.3%

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

   4. Taylor expanded in x around 0 43.0%

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

   5. Taylor expanded in x around 0 49.3%

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

   6. Taylor expanded in eps around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. distribute-lft-neg-out 73.4%

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

      3. *-commutative 73.4%

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

   8. Simplified 73.4%

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

   if 1.2500000000000001e-183 < x < 1.35e-114 or 6.0000000000000004e-66 < x < 45

   1. Initial program 75.8%

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

      1. sub-neg 75.8%

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

      2. neg-sub0 75.8%

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

      3. associate-+r- 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in x around 0 47.4%

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

   5. Taylor expanded in x around 0 23.2%

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

      1. *-commutative 23.2%

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

      2. flip-+ 50.9%

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

      3. associate-*r/ 50.9%

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

      4. metadata-eval 50.9%

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

      5. add-sqr-sqrt 42.7%

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

      6. sqrt-unprod 22.0%

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

      7. sqr-neg 22.0%

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

      8. sqrt-unprod 29.4%

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

      9. add-sqr-sqrt 42.1%

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

      10. neg-sub0 42.1%

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

      11. metadata-eval 42.1%

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

      12. associate--r- 42.1%

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

      13. metadata-eval 42.1%

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

      14. metadata-eval 42.1%

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

   7. Applied egg-rr 42.1%

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

      1. associate-/l* 42.1%

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

      2. +-commutative 42.1%

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

   9. Simplified 42.1%

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

   if 1.35e-114 < x < 6.0000000000000004e-66

   1. Initial program 54.4%

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

      1. sub-neg 54.4%

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

      2. neg-sub0 54.4%

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

      3. associate-+r- 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in x around 0 31.6%

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

   5. Taylor expanded in eps around inf 77.2%

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

      1. exp-prod 77.2%

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

      2. *-commutative 77.2%

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

      3. sub-neg 77.2%

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

      4. neg-mul-1 77.2%

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

      5. *-commutative 77.2%

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

      6. exp-prod 77.2%

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

      7. +-commutative 77.2%

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

      8. associate-*r* 77.2%

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

      9. neg-mul-1 77.2%

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

      10. sub-neg 77.2%

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

      11. *-commutative 77.2%

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

      12. neg-mul-1 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in eps around inf 77.2%

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

   9. Taylor expanded in eps around 0 70.9%

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

   if 45 < x < 3.9000000000000002e170

   1. Initial program 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in eps around 0 59.8%

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

      1. rec-exp 59.8%

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

      2. div-sub 59.8%

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

      3. rec-exp 59.8%

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

      4. neg-mul-1 59.8%

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

      5. rec-exp 59.8%

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

      6. neg-mul-1 59.8%

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

      7. +-inverses 59.8%

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

   6. Simplified 59.8%

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

   if 3.9000000000000002e170 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 24.4%

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

   5. Taylor expanded in x around 0 19.4%

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

      1. flip-+ 19.4%

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

      2. associate-*l/ 22.8%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{\left(1 \cdot 1 - \varepsilon \cdot \varepsilon\right) \cdot x}{1 - \varepsilon}}}{2} \]

      3. metadata-eval 22.8%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(\color{blue}{1} - \varepsilon \cdot \varepsilon\right) \cdot x}{1 - \varepsilon}}{2} \]

      4. add-sqr-sqrt 22.8%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{\sqrt{1 - \varepsilon} \cdot \sqrt{1 - \varepsilon}}}}{2} \]

      5. sqrt-unprod 31.0%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{\sqrt{\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}}}}{2} \]

      6. sqr-neg 31.0%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\sqrt{\color{blue}{\left(-\left(1 - \varepsilon\right)\right) \cdot \left(-\left(1 - \varepsilon\right)\right)}}}}{2} \]

      7. sqrt-unprod 34.2%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{\sqrt{-\left(1 - \varepsilon\right)} \cdot \sqrt{-\left(1 - \varepsilon\right)}}}}{2} \]

      8. add-sqr-sqrt 34.7%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{-\left(1 - \varepsilon\right)}}}{2} \]

      9. neg-sub0 34.7%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{0 - \left(1 - \varepsilon\right)}}}{2} \]

      10. metadata-eval 34.7%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{\log 1} - \left(1 - \varepsilon\right)}}{2} \]

      11. associate--r- 34.7%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{\left(\log 1 - 1\right) + \varepsilon}}}{2} \]

      12. metadata-eval 34.7%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\left(\color{blue}{0} - 1\right) + \varepsilon}}{2} \]

      13. metadata-eval 34.7%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{\color{blue}{-1} + \varepsilon}}{2} \]

   7. Applied egg-rr 34.7%

      \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{\left(1 - \varepsilon \cdot \varepsilon\right) \cdot x}{-1 + \varepsilon}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-183}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-114}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-66}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 45:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\varepsilon + -1}}{2}\\ \end{array} \]

Alternative 11: 53.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 + \varepsilon \cdot x}{2}\\ t_1 := \frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{if}\;x \leq 1.55 \cdot 10^{-179}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 31.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 2.0 (* eps x)) 2.0))
        (t_1
         (/
          (+
           2.0
           (*
            (- -1.0 (/ -1.0 eps))
            (/ x (/ (+ eps -1.0) (- 1.0 (* eps eps))))))
          2.0)))
   (if (<= x 1.55e-179)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x 9.5e-114)
       t_1
       (if (<= x 5e-62)
         t_0
         (if (<= x 31.5) t_1 (if (<= x 7.2e+170) 0.0 t_0)))))))

C

double code(double x, double eps) {
	double t_0 = (2.0 + (eps * x)) / 2.0;
	double t_1 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	double tmp;
	if (x <= 1.55e-179) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 9.5e-114) {
		tmp = t_1;
	} else if (x <= 5e-62) {
		tmp = t_0;
	} else if (x <= 31.5) {
		tmp = t_1;
	} else if (x <= 7.2e+170) {
		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 = (2.0d0 + (eps * x)) / 2.0d0
    t_1 = (2.0d0 + (((-1.0d0) - ((-1.0d0) / eps)) * (x / ((eps + (-1.0d0)) / (1.0d0 - (eps * eps)))))) / 2.0d0
    if (x <= 1.55d-179) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 9.5d-114) then
        tmp = t_1
    else if (x <= 5d-62) then
        tmp = t_0
    else if (x <= 31.5d0) then
        tmp = t_1
    else if (x <= 7.2d+170) 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 = (2.0 + (eps * x)) / 2.0;
	double t_1 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	double tmp;
	if (x <= 1.55e-179) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 9.5e-114) {
		tmp = t_1;
	} else if (x <= 5e-62) {
		tmp = t_0;
	} else if (x <= 31.5) {
		tmp = t_1;
	} else if (x <= 7.2e+170) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (2.0 + (eps * x)) / 2.0
	t_1 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0
	tmp = 0
	if x <= 1.55e-179:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 9.5e-114:
		tmp = t_1
	elif x <= 5e-62:
		tmp = t_0
	elif x <= 31.5:
		tmp = t_1
	elif x <= 7.2e+170:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(2.0 + Float64(eps * x)) / 2.0)
	t_1 = Float64(Float64(2.0 + Float64(Float64(-1.0 - Float64(-1.0 / eps)) * Float64(x / Float64(Float64(eps + -1.0) / Float64(1.0 - Float64(eps * eps)))))) / 2.0)
	tmp = 0.0
	if (x <= 1.55e-179)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 9.5e-114)
		tmp = t_1;
	elseif (x <= 5e-62)
		tmp = t_0;
	elseif (x <= 31.5)
		tmp = t_1;
	elseif (x <= 7.2e+170)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (2.0 + (eps * x)) / 2.0;
	t_1 = (2.0 + ((-1.0 - (-1.0 / eps)) * (x / ((eps + -1.0) / (1.0 - (eps * eps)))))) / 2.0;
	tmp = 0.0;
	if (x <= 1.55e-179)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 9.5e-114)
		tmp = t_1;
	elseif (x <= 5e-62)
		tmp = t_0;
	elseif (x <= 31.5)
		tmp = t_1;
	elseif (x <= 7.2e+170)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 + N[(N[(-1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(eps + -1.0), $MachinePrecision] / N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, 1.55e-179], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.5e-114], t$95$1, If[LessEqual[x, 5e-62], t$95$0, If[LessEqual[x, 31.5], t$95$1, If[LessEqual[x, 7.2e+170], 0.0, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.5500000000000001e-179

   1. Initial program 58.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 58.3%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 58.3%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 58.3%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 43.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. Taylor expanded in x around 0 49.3%

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]

   6. Taylor expanded in eps around inf 73.4%

      \[\leadsto \frac{2 + \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 73.4%

         \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon \cdot x\right)}}{2} \]

      2. distribute-lft-neg-out 73.4%

         \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon\right) \cdot x}}{2} \]

      3. *-commutative 73.4%

         \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-\varepsilon\right)}}{2} \]

   8. Simplified 73.4%

      \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-\varepsilon\right)}}{2} \]

   if 1.5500000000000001e-179 < x < 9.49999999999999958e-114 or 5.0000000000000002e-62 < x < 31.5

   1. Initial program 75.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 75.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 75.8%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 75.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 47.4%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. Taylor expanded in x around 0 23.2%

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]
   6. Step-by-step derivation

      1. *-commutative 23.2%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)}}{2} \]

      2. flip-+ 50.9%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(x \cdot \color{blue}{\frac{1 \cdot 1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon}}\right)}{2} \]

      3. associate-*r/ 50.9%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{x \cdot \left(1 \cdot 1 - \varepsilon \cdot \varepsilon\right)}{1 - \varepsilon}}}{2} \]

      4. metadata-eval 50.9%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(\color{blue}{1} - \varepsilon \cdot \varepsilon\right)}{1 - \varepsilon}}{2} \]

      5. add-sqr-sqrt 42.7%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{\sqrt{1 - \varepsilon} \cdot \sqrt{1 - \varepsilon}}}}{2} \]

      6. sqrt-unprod 22.0%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{\sqrt{\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)}}}}{2} \]

      7. sqr-neg 22.0%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\sqrt{\color{blue}{\left(-\left(1 - \varepsilon\right)\right) \cdot \left(-\left(1 - \varepsilon\right)\right)}}}}{2} \]

      8. sqrt-unprod 29.4%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{\sqrt{-\left(1 - \varepsilon\right)} \cdot \sqrt{-\left(1 - \varepsilon\right)}}}}{2} \]

      9. add-sqr-sqrt 42.1%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{-\left(1 - \varepsilon\right)}}}{2} \]

      10. neg-sub0 42.1%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{0 - \left(1 - \varepsilon\right)}}}{2} \]

      11. metadata-eval 42.1%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{\log 1} - \left(1 - \varepsilon\right)}}{2} \]

      12. associate--r- 42.1%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{\left(\log 1 - 1\right) + \varepsilon}}}{2} \]

      13. metadata-eval 42.1%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\left(\color{blue}{0} - 1\right) + \varepsilon}}{2} \]

      14. metadata-eval 42.1%

          \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{\color{blue}{-1} + \varepsilon}}{2} \]

   7. Applied egg-rr 42.1%

      \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{x \cdot \left(1 - \varepsilon \cdot \varepsilon\right)}{-1 + \varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* 42.1%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{x}{\frac{-1 + \varepsilon}{1 - \varepsilon \cdot \varepsilon}}}}{2} \]

      2. +-commutative 42.1%

         \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{x}{\frac{\color{blue}{\varepsilon + -1}}{1 - \varepsilon \cdot \varepsilon}}}{2} \]

   9. Simplified 42.1%

      \[\leadsto \frac{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}}{2} \]

   if 9.49999999999999958e-114 < x < 5.0000000000000002e-62 or 7.1999999999999999e170 < x

   1. Initial program 82.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 82.4%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 82.4%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 82.4%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 82.4%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 40.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   5. Taylor expanded in eps around inf 58.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2} \]
   6. Step-by-step derivation

      1. exp-prod 58.2%

         \[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      2. *-commutative 58.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right)}} + 1}{2} \]

      3. sub-neg 58.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)} + 1}{2} \]

      4. neg-mul-1 58.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)} + 1}{2} \]

      5. *-commutative 58.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      6. exp-prod 58.2%

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      7. +-commutative 58.2%

         \[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2} \]

      8. associate-*r* 58.2%

         \[\leadsto \frac{1 + e^{\color{blue}{\left(-1 \cdot \left(1 + -1 \cdot \varepsilon\right)\right) \cdot x}}}{2} \]

      9. neg-mul-1 58.2%

         \[\leadsto \frac{1 + e^{\left(-1 \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot x}}{2} \]

      10. sub-neg 58.2%

          \[\leadsto \frac{1 + e^{\left(-1 \cdot \color{blue}{\left(1 - \varepsilon\right)}\right) \cdot x}}{2} \]

      11. *-commutative 58.2%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 - \varepsilon\right)\right)}}}{2} \]

      12. neg-mul-1 58.2%

          \[\leadsto \frac{1 + e^{x \cdot \color{blue}{\left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   7. Simplified 58.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   8. Taylor expanded in eps around inf 58.1%

      \[\leadsto \frac{1 + e^{\color{blue}{\varepsilon \cdot x}}}{2} \]

   9. Taylor expanded in eps around 0 46.9%

      \[\leadsto \frac{\color{blue}{2 + \varepsilon \cdot x}}{2} \]

   if 31.5 < x < 7.1999999999999999e170

   1. Initial program 95.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 59.8%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. rec-exp 59.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. div-sub 59.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{e^{x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      3. rec-exp 59.8%

         \[\leadsto \frac{\frac{\color{blue}{e^{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      4. neg-mul-1 59.8%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      5. rec-exp 59.8%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      6. neg-mul-1 59.8%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      7. +-inverses 59.8%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 59.8%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-179}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-62}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 31.5:\\ \;\;\;\;\frac{2 + \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \frac{x}{\frac{\varepsilon + -1}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 12: 53.2% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 2.0 x) 2.0)
   (if (<= x 9.6e+170) 0.0 (/ (+ 2.0 (* eps x)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 9.6e+170) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (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 <= 2.0d0) then
        tmp = (2.0d0 - x) / 2.0d0
    else if (x <= 9.6d+170) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 9.6e+170) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	elif x <= 9.6e+170:
		tmp = 0.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	elseif (x <= 9.6e+170)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	elseif (x <= 9.6e+170)
		tmp = 0.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.6e+170], 0.0, N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 60.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 60.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 60.2%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 60.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 39.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   5. Taylor expanded in eps around inf 79.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2} \]
   6. Step-by-step derivation

      1. exp-prod 79.2%

         \[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      2. *-commutative 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right)}} + 1}{2} \]

      3. sub-neg 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)} + 1}{2} \]

      4. neg-mul-1 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)} + 1}{2} \]

      5. *-commutative 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      6. exp-prod 79.2%

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      7. +-commutative 79.2%

         \[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2} \]

      8. associate-*r* 79.2%

         \[\leadsto \frac{1 + e^{\color{blue}{\left(-1 \cdot \left(1 + -1 \cdot \varepsilon\right)\right) \cdot x}}}{2} \]

      9. neg-mul-1 79.2%

         \[\leadsto \frac{1 + e^{\left(-1 \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot x}}{2} \]

      10. sub-neg 79.2%

          \[\leadsto \frac{1 + e^{\left(-1 \cdot \color{blue}{\left(1 - \varepsilon\right)}\right) \cdot x}}{2} \]

      11. *-commutative 79.2%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 - \varepsilon\right)\right)}}}{2} \]

      12. neg-mul-1 79.2%

          \[\leadsto \frac{1 + e^{x \cdot \color{blue}{\left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   7. Simplified 79.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   8. Taylor expanded in x around 0 67.2%

      \[\leadsto \frac{\color{blue}{2 + \left(\varepsilon - 1\right) \cdot x}}{2} \]

   9. Taylor expanded in eps around 0 64.3%

      \[\leadsto \frac{2 + \color{blue}{-1 \cdot x}}{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 64.3%

          \[\leadsto \frac{2 + \color{blue}{\left(-x\right)}}{2} \]

   11. Simplified 64.3%

       \[\leadsto \frac{2 + \color{blue}{\left(-x\right)}}{2} \]

   if 2 < x < 9.5999999999999999e170

   1. Initial program 95.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 59.8%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. rec-exp 59.8%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. div-sub 59.8%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{e^{x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      3. rec-exp 59.8%

         \[\leadsto \frac{\frac{\color{blue}{e^{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      4. neg-mul-1 59.8%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      5. rec-exp 59.8%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      6. neg-mul-1 59.8%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      7. +-inverses 59.8%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 59.8%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 9.5999999999999999e170 < 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. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   5. Taylor expanded in eps around inf 46.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2} \]
   6. Step-by-step derivation

      1. exp-prod 46.2%

         \[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      2. *-commutative 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right)}} + 1}{2} \]

      3. sub-neg 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)} + 1}{2} \]

      4. neg-mul-1 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)} + 1}{2} \]

      5. *-commutative 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      6. exp-prod 46.2%

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      7. +-commutative 46.2%

         \[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2} \]

      8. associate-*r* 46.2%

         \[\leadsto \frac{1 + e^{\color{blue}{\left(-1 \cdot \left(1 + -1 \cdot \varepsilon\right)\right) \cdot x}}}{2} \]

      9. neg-mul-1 46.2%

         \[\leadsto \frac{1 + e^{\left(-1 \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot x}}{2} \]

      10. sub-neg 46.2%

          \[\leadsto \frac{1 + e^{\left(-1 \cdot \color{blue}{\left(1 - \varepsilon\right)}\right) \cdot x}}{2} \]

      11. *-commutative 46.2%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 - \varepsilon\right)\right)}}}{2} \]

      12. neg-mul-1 46.2%

          \[\leadsto \frac{1 + e^{x \cdot \color{blue}{\left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   7. Simplified 46.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   8. Taylor expanded in eps around inf 46.1%

      \[\leadsto \frac{1 + e^{\color{blue}{\varepsilon \cdot x}}}{2} \]

   9. Taylor expanded in eps around 0 31.8%

      \[\leadsto \frac{\color{blue}{2 + \varepsilon \cdot x}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 13: 56.3% accurate, 20.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 230.0)
   (/ (- 2.0 (* eps x)) 2.0)
   (if (<= x 4.2e+170) 0.0 (/ (+ 2.0 (* eps x)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 230.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 4.2e+170) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (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 <= 230.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 4.2d+170) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 230.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 4.2e+170) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 230.0:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 4.2e+170:
		tmp = 0.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 230.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 4.2e+170)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 230.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 4.2e+170)
		tmp = 0.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 230.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2e+170], 0.0, N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 230

   1. Initial program 59.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 59.6%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 59.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 59.6%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 43.6%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. Taylor expanded in x around 0 43.3%

      \[\leadsto \frac{\color{blue}{2 + \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}{2} \]

   6. Taylor expanded in eps around inf 67.6%

      \[\leadsto \frac{2 + \color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
   7. Step-by-step derivation

      1. mul-1-neg 67.6%

         \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon \cdot x\right)}}{2} \]

      2. distribute-lft-neg-out 67.6%

         \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon\right) \cdot x}}{2} \]

      3. *-commutative 67.6%

         \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-\varepsilon\right)}}{2} \]

   8. Simplified 67.6%

      \[\leadsto \frac{2 + \color{blue}{x \cdot \left(-\varepsilon\right)}}{2} \]

   if 230 < x < 4.19999999999999996e170

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 62.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. rec-exp 62.5%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. div-sub 62.5%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{e^{x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      3. rec-exp 62.5%

         \[\leadsto \frac{\frac{\color{blue}{e^{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      4. neg-mul-1 62.5%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      5. rec-exp 62.5%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      6. neg-mul-1 62.5%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      7. +-inverses 62.5%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 62.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 4.19999999999999996e170 < 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. sub-neg 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 100.0%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 100.0%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   5. Taylor expanded in eps around inf 46.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2} \]
   6. Step-by-step derivation

      1. exp-prod 46.2%

         \[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      2. *-commutative 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right)}} + 1}{2} \]

      3. sub-neg 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)} + 1}{2} \]

      4. neg-mul-1 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)} + 1}{2} \]

      5. *-commutative 46.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      6. exp-prod 46.2%

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      7. +-commutative 46.2%

         \[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2} \]

      8. associate-*r* 46.2%

         \[\leadsto \frac{1 + e^{\color{blue}{\left(-1 \cdot \left(1 + -1 \cdot \varepsilon\right)\right) \cdot x}}}{2} \]

      9. neg-mul-1 46.2%

         \[\leadsto \frac{1 + e^{\left(-1 \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot x}}{2} \]

      10. sub-neg 46.2%

          \[\leadsto \frac{1 + e^{\left(-1 \cdot \color{blue}{\left(1 - \varepsilon\right)}\right) \cdot x}}{2} \]

      11. *-commutative 46.2%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 - \varepsilon\right)\right)}}}{2} \]

      12. neg-mul-1 46.2%

          \[\leadsto \frac{1 + e^{x \cdot \color{blue}{\left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   7. Simplified 46.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   8. Taylor expanded in eps around inf 46.1%

      \[\leadsto \frac{1 + e^{\color{blue}{\varepsilon \cdot x}}}{2} \]

   9. Taylor expanded in eps around 0 31.8%

      \[\leadsto \frac{\color{blue}{2 + \varepsilon \cdot x}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+170}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 14: 56.4% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 2.0) (/ (- 2.0 x) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - 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 <= 2.0d0) then
        tmp = (2.0d0 - 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 <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 60.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 60.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 60.2%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 60.2%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 60.2%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in x around 0 39.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   5. Taylor expanded in eps around inf 79.2%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + 1}}{2} \]
   6. Step-by-step derivation

      1. exp-prod 79.2%

         \[\leadsto \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 - \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      2. *-commutative 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(x \cdot \left(1 - \varepsilon\right)\right)}} + 1}{2} \]

      3. sub-neg 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right)} + 1}{2} \]

      4. neg-mul-1 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\left(x \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right)} + 1}{2} \]

      5. *-commutative 79.2%

         \[\leadsto \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      6. exp-prod 79.2%

         \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} + 1}{2} \]

      7. +-commutative 79.2%

         \[\leadsto \frac{\color{blue}{1 + e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}}}{2} \]

      8. associate-*r* 79.2%

         \[\leadsto \frac{1 + e^{\color{blue}{\left(-1 \cdot \left(1 + -1 \cdot \varepsilon\right)\right) \cdot x}}}{2} \]

      9. neg-mul-1 79.2%

         \[\leadsto \frac{1 + e^{\left(-1 \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot x}}{2} \]

      10. sub-neg 79.2%

          \[\leadsto \frac{1 + e^{\left(-1 \cdot \color{blue}{\left(1 - \varepsilon\right)}\right) \cdot x}}{2} \]

      11. *-commutative 79.2%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 - \varepsilon\right)\right)}}}{2} \]

      12. neg-mul-1 79.2%

          \[\leadsto \frac{1 + e^{x \cdot \color{blue}{\left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   7. Simplified 79.2%

      \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-\left(1 - \varepsilon\right)\right)}}}{2} \]

   8. Taylor expanded in x around 0 67.2%

      \[\leadsto \frac{\color{blue}{2 + \left(\varepsilon - 1\right) \cdot x}}{2} \]

   9. Taylor expanded in eps around 0 64.3%

      \[\leadsto \frac{2 + \color{blue}{-1 \cdot x}}{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 64.3%

          \[\leadsto \frac{2 + \color{blue}{\left(-x\right)}}{2} \]

   11. Simplified 64.3%

       \[\leadsto \frac{2 + \color{blue}{\left(-x\right)}}{2} \]

   if 2 < x

   1. Initial program 97.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 97.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 50.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. rec-exp 50.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. div-sub 50.1%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{e^{x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      3. rec-exp 50.2%

         \[\leadsto \frac{\frac{\color{blue}{e^{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      4. neg-mul-1 50.2%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      5. rec-exp 50.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      6. neg-mul-1 50.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      7. +-inverses 50.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 50.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 56.4% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1800:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 1800.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1800.0) {
		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 <= 1800.0d0) 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 <= 1800.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1800.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1800.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1800.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1800.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1800

   1. Initial program 59.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Step-by-step derivation

      1. sub-neg 59.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      2. neg-sub0 59.8%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

      3. associate-+r- 59.8%

         \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

   4. Taylor expanded in eps around 0 63.9%

      \[\leadsto \frac{\color{blue}{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}}{2} \]
   5. Step-by-step derivation

      1. neg-mul-1 63.9%

         \[\leadsto \frac{\left(e^{-1 \cdot x} \cdot x + e^{\color{blue}{-x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      2. rec-exp 63.9%

         \[\leadsto \frac{\left(e^{-1 \cdot x} \cdot x + \color{blue}{\frac{1}{e^{x}}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      3. *-commutative 63.9%

         \[\leadsto \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + \frac{1}{e^{x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      4. neg-mul-1 63.9%

         \[\leadsto \frac{\left(x \cdot e^{\color{blue}{-x}} + \frac{1}{e^{x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      5. rec-exp 63.9%

         \[\leadsto \frac{\left(x \cdot \color{blue}{\frac{1}{e^{x}}} + \frac{1}{e^{x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      6. distribute-lft1-in 63.9%

         \[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot \frac{1}{e^{x}}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      7. rec-exp 63.9%

         \[\leadsto \frac{\left(x + 1\right) \cdot \color{blue}{e^{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

      8. distribute-lft-out 63.9%

         \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2} \]

      9. mul-1-neg 63.9%

         \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2} \]

      10. neg-mul-1 63.9%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(e^{-1 \cdot x} \cdot x + e^{\color{blue}{-x}}\right)\right)}{2} \]

      11. rec-exp 63.9%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(e^{-1 \cdot x} \cdot x + \color{blue}{\frac{1}{e^{x}}}\right)\right)}{2} \]

      12. *-commutative 63.9%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + \frac{1}{e^{x}}\right)\right)}{2} \]

      13. neg-mul-1 63.9%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x \cdot e^{\color{blue}{-x}} + \frac{1}{e^{x}}\right)\right)}{2} \]

      14. rec-exp 63.9%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x \cdot \color{blue}{\frac{1}{e^{x}}} + \frac{1}{e^{x}}\right)\right)}{2} \]

      15. distribute-lft1-in 64.0%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot \frac{1}{e^{x}}}\right)}{2} \]

      16. rec-exp 64.0%

          \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot \color{blue}{e^{-x}}\right)}{2} \]

   6. Simplified 64.0%

      \[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}}{2} \]

   7. Taylor expanded in x around 0 62.8%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
   8. Step-by-step derivation

      1. unpow2 62.8%

         \[\leadsto \frac{\left(1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   9. Simplified 62.8%

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   10. Taylor expanded in x around 0 63.2%

       \[\leadsto \color{blue}{1} \]

   if 1800 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\varepsilon \cdot x - x}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]

   4. Taylor expanded in eps around 0 52.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}}}{2} \]
   5. Step-by-step derivation

      1. rec-exp 52.2%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]

      2. div-sub 52.2%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{e^{x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      3. rec-exp 52.2%

         \[\leadsto \frac{\frac{\color{blue}{e^{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      4. neg-mul-1 52.2%

         \[\leadsto \frac{\frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      5. rec-exp 52.2%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]

      6. neg-mul-1 52.2%

         \[\leadsto \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]

      7. +-inverses 52.2%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 52.2%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1800:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 43.3% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 70.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Step-by-step derivation

   1. sub-neg 70.5%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \left(-\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

   2. neg-sub0 70.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + \color{blue}{\left(0 - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \]

   3. associate-+r- 70.5%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \]

3. Simplified 70.5%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]

4. Taylor expanded in eps around 0 60.8%

   \[\leadsto \frac{\color{blue}{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}}{2} \]
5. Step-by-step derivation

   1. neg-mul-1 60.8%

      \[\leadsto \frac{\left(e^{-1 \cdot x} \cdot x + e^{\color{blue}{-x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   2. rec-exp 60.8%

      \[\leadsto \frac{\left(e^{-1 \cdot x} \cdot x + \color{blue}{\frac{1}{e^{x}}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   3. *-commutative 60.8%

      \[\leadsto \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + \frac{1}{e^{x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   4. neg-mul-1 60.8%

      \[\leadsto \frac{\left(x \cdot e^{\color{blue}{-x}} + \frac{1}{e^{x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   5. rec-exp 60.8%

      \[\leadsto \frac{\left(x \cdot \color{blue}{\frac{1}{e^{x}}} + \frac{1}{e^{x}}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   6. distribute-lft1-in 60.8%

      \[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot \frac{1}{e^{x}}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   7. rec-exp 60.8%

      \[\leadsto \frac{\left(x + 1\right) \cdot \color{blue}{e^{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]

   8. distribute-lft-out 60.8%

      \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2} \]

   9. mul-1-neg 60.8%

      \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2} \]

   10. neg-mul-1 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(e^{-1 \cdot x} \cdot x + e^{\color{blue}{-x}}\right)\right)}{2} \]

   11. rec-exp 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(e^{-1 \cdot x} \cdot x + \color{blue}{\frac{1}{e^{x}}}\right)\right)}{2} \]

   12. *-commutative 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + \frac{1}{e^{x}}\right)\right)}{2} \]

   13. neg-mul-1 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x \cdot e^{\color{blue}{-x}} + \frac{1}{e^{x}}\right)\right)}{2} \]

   14. rec-exp 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x \cdot \color{blue}{\frac{1}{e^{x}}} + \frac{1}{e^{x}}\right)\right)}{2} \]

   15. distribute-lft1-in 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot \frac{1}{e^{x}}}\right)}{2} \]

   16. rec-exp 60.8%

       \[\leadsto \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot \color{blue}{e^{-x}}\right)}{2} \]

6. Simplified 60.8%

   \[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}}{2} \]

7. Taylor expanded in x around 0 46.4%

   \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
8. Step-by-step derivation

   1. unpow2 46.4%

      \[\leadsto \frac{\left(1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

9. Simplified 46.4%

   \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

10. Taylor expanded in x around 0 47.2%

    \[\leadsto \color{blue}{1} \]

11. Final simplification 47.2%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023274 
(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))