NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.2%

Time: 21.4s

Alternatives: 23

Speedup: 10.8×

• 38.6% 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: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\log \left(\mathsf{fma}\left(t\_0, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right) \cdot 0.3333333333333333}\right)}^{3}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<=
        (+
         (* t_0 (exp (* x (+ eps -1.0))))
         (* (exp (* x (- -1.0 eps))) (- 1.0 (/ 1.0 eps))))
        4.0)
     (/ (* 2.0 (exp (- (log1p x) x))) 2.0)
     (/
      (pow
       (exp
        (*
         (log
          (fma
           t_0
           (exp (fma eps x x))
           (exp (- (log1p (/ 1.0 eps)) (fma eps x x)))))
         0.3333333333333333))
       3.0)
      2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) <= 4.0) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else {
		tmp = pow(exp((log(fma(t_0, exp(fma(eps, x, x)), exp((log1p((1.0 / eps)) - fma(eps, x, x))))) * 0.3333333333333333)), 3.0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 - Float64(1.0 / eps)))) <= 4.0)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	else
		tmp = Float64((exp(Float64(log(fma(t_0, exp(fma(eps, x, x)), exp(Float64(log1p(Float64(1.0 / eps)) - fma(eps, x, x))))) * 0.3333333333333333)) ^ 3.0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Exp[N[(N[Log[N[(t$95$0 * N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[Log[1 + N[(1.0 / eps), $MachinePrecision]], $MachinePrecision] - N[(eps * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 54.9%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in eps around 0 54.3%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-sum 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   10. Simplified 100.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.9%

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

   3. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 82.2%

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

      2. pow3 82.2%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]
   7. Step-by-step derivation

      1. pow1/3 99.9%

         \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}^{0.3333333333333333}\right)}}^{3}}{2} \]

      2. pow-to-exp 99.9%

         \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right) \cdot 0.3333333333333333}\right)}}^{3}}{2} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right) \cdot 0.3333333333333333}\right)}}^{3}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(e^{\log \left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right) \cdot 0.3333333333333333}\right)}^{3}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(t\_0, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}^{0.3333333333333333}\right)}^{3}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<=
        (+
         (* t_0 (exp (* x (+ eps -1.0))))
         (* (exp (* x (- -1.0 eps))) (- 1.0 (/ 1.0 eps))))
        4.0)
     (/ (* 2.0 (exp (- (log1p x) x))) 2.0)
     (/
      (pow
       (pow
        (fma
         t_0
         (exp (fma eps x x))
         (exp (- (log1p (/ 1.0 eps)) (fma eps x x))))
        0.3333333333333333)
       3.0)
      2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) <= 4.0) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else {
		tmp = pow(pow(fma(t_0, exp(fma(eps, x, x)), exp((log1p((1.0 / eps)) - fma(eps, x, x)))), 0.3333333333333333), 3.0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 - Float64(1.0 / eps)))) <= 4.0)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	else
		tmp = Float64(((fma(t_0, exp(fma(eps, x, x)), exp(Float64(log1p(Float64(1.0 / eps)) - fma(eps, x, x)))) ^ 0.3333333333333333) ^ 3.0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(t$95$0 * N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[Log[1 + N[(1.0 / eps), $MachinePrecision]], $MachinePrecision] - N[(eps * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 54.9%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in eps around 0 54.3%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-sum 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   10. Simplified 100.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.9%

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

   3. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 82.2%

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

      2. pow3 82.2%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]
   7. Step-by-step derivation

      1. pow1/3 99.9%

         \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}^{0.3333333333333333}\right)}}^{3}}{2} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{{\color{blue}{\left({\left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}^{0.3333333333333333}\right)}}^{3}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left({\left(\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}^{0.3333333333333333}\right)}^{3}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\mathsf{fma}\left(t\_0, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<=
        (+
         (* t_0 (exp (* x (+ eps -1.0))))
         (* (exp (* x (- -1.0 eps))) (- 1.0 (/ 1.0 eps))))
        4.0)
     (/ (* 2.0 (exp (- (log1p x) x))) 2.0)
     (/
      (pow
       (cbrt
        (fma
         t_0
         (exp (fma eps x x))
         (exp (- (log1p (/ 1.0 eps)) (fma eps x x)))))
       3.0)
      2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))) <= 4.0) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else {
		tmp = pow(cbrt(fma(t_0, exp(fma(eps, x, x)), exp((log1p((1.0 / eps)) - fma(eps, x, x))))), 3.0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 - Float64(1.0 / eps)))) <= 4.0)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	else
		tmp = Float64((cbrt(fma(t_0, exp(fma(eps, x, x)), exp(Float64(log1p(Float64(1.0 / eps)) - fma(eps, x, x))))) ^ 3.0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Power[N[Power[N[(t$95$0 * N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[Log[1 + N[(1.0 / eps), $MachinePrecision]], $MachinePrecision] - N[(eps * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 54.9%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in eps around 0 54.3%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-sum 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   10. Simplified 100.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.9%

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

   3. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cube-cbrt 82.2%

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

      2. pow3 82.2%

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

   6. Applied egg-rr 99.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{1}{\varepsilon}\\ t_1 := 1 + \frac{1}{\varepsilon}\\ t_2 := e^{x \cdot \left(\varepsilon + -1\right)}\\ \mathbf{if}\;t\_1 \cdot t\_2 + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot t\_0 \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, t\_2, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot t\_0\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 1.0 eps)))
        (t_1 (+ 1.0 (/ 1.0 eps)))
        (t_2 (exp (* x (+ eps -1.0)))))
   (if (<= (+ (* t_1 t_2) (* (exp (* x (- -1.0 eps))) t_0)) 4.0)
     (/ (* 2.0 (exp (- (log1p x) x))) 2.0)
     (/ (fma t_1 t_2 (* (pow (exp (+ 1.0 eps)) (- x)) t_0)) 2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (1.0 / eps);
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = exp((x * (eps + -1.0)));
	double tmp;
	if (((t_1 * t_2) + (exp((x * (-1.0 - eps))) * t_0)) <= 4.0) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else {
		tmp = fma(t_1, t_2, (pow(exp((1.0 + eps)), -x) * t_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(1.0 / eps))
	t_2 = exp(Float64(x * Float64(eps + -1.0)))
	tmp = 0.0
	if (Float64(Float64(t_1 * t_2) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * t_0)) <= 4.0)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	else
		tmp = Float64(fma(t_1, t_2, Float64((exp(Float64(1.0 + eps)) ^ Float64(-x)) * t_0)) / 2.0);
	end
	return 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[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * t$95$2), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 * t$95$2 + N[(N[Power[N[Exp[N[(1.0 + eps), $MachinePrecision]], $MachinePrecision], (-x)], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 54.9%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in eps around 0 54.3%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-sum 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   10. Simplified 100.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)\\ \mathbf{if}\;t\_0 \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (+
          (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
          (* (exp (* x (- -1.0 eps))) (- 1.0 (/ 1.0 eps))))))
   (if (<= t_0 4.0) (/ (* 2.0 (exp (- (log1p x) 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)))) + (exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)));
	double tmp;
	if (t_0 <= 4.0) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)));
	double tmp;
	if (t_0 <= 4.0) {
		tmp = (2.0 * Math.exp((Math.log1p(x) - 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)))) + (math.exp((x * (-1.0 - eps))) * (1.0 - (1.0 / eps)))
	tmp = 0
	if t_0 <= 4.0:
		tmp = (2.0 * math.exp((math.log1p(x) - 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(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 - Float64(1.0 / eps))))
	tmp = 0.0
	if (t_0 <= 4.0)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return 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[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4.0], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - 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 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 54.9%

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

   3. Simplified 27.0%

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

   5. Taylor expanded in eps around 0 54.3%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-in 100.0%

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

      2. associate-*r* 100.0%

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

      3. distribute-lft-out 100.0%

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

      4. distribute-rgt1-in 100.0%

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

      5. rem-exp-log 100.0%

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

      6. exp-sum 100.0%

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

      7. unsub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. log1p-define 100.0%

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

   10. Simplified 100.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 4:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {e}^{\left(x \cdot \left(1 + \varepsilon\right)\right)} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.12 \cdot 10^{+94}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2900000000.0)
   (/ (* 2.0 (exp (- (log1p x) x))) 2.0)
   (if (<= eps 2.45e+81)
     (/
      (+ (* (+ 1.0 (/ 1.0 eps)) (pow E (* x (+ 1.0 eps)))) (- 1.0 (/ 1.0 eps)))
      2.0)
     (if (<= eps 1.12e+94)
       (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
       (/
        (+
         2.0
         (* x (+ (- -1.0 eps) (* 0.5 (+ x (* eps (+ (* x 2.0) (* eps x))))))))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else if (eps <= 2.45e+81) {
		tmp = (((1.0 + (1.0 / eps)) * pow(((double) M_E), (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (eps <= 1.12e+94) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (2.0 * Math.exp((Math.log1p(x) - x))) / 2.0;
	} else if (eps <= 2.45e+81) {
		tmp = (((1.0 + (1.0 / eps)) * Math.pow(Math.E, (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (eps <= 1.12e+94) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2900000000.0:
		tmp = (2.0 * math.exp((math.log1p(x) - x))) / 2.0
	elif eps <= 2.45e+81:
		tmp = (((1.0 + (1.0 / eps)) * math.pow(math.e, (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0
	elif eps <= 1.12e+94:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2900000000.0)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	elseif (eps <= 2.45e+81)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * (exp(1) ^ Float64(x * Float64(1.0 + eps)))) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (eps <= 1.12e+94)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x + Float64(eps * Float64(Float64(x * 2.0) + Float64(eps * x)))))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2900000000.0], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.45e+81], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Power[E, N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.12e+94], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x + N[(eps * N[(N[(x * 2.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 2.9e9

   1. Initial program 57.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in eps around 0 38.0%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in eps around 0 81.3%

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

      1. distribute-rgt-in 81.3%

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

      2. associate-*r* 81.3%

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

      3. distribute-lft-out 81.3%

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

      4. distribute-rgt1-in 81.3%

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

      5. rem-exp-log 81.3%

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

      6. exp-sum 81.3%

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

      7. unsub-neg 81.3%

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

      8. +-commutative 81.3%

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

      9. log1p-define 81.3%

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

   10. Simplified 81.3%

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

   if 2.9e9 < eps < 2.45000000000000011e81

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. exp-prod 82.5%

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

      3. *-commutative 82.5%

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

      4. add-sqr-sqrt 30.0%

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

      5. sqrt-unprod 43.4%

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

      6. sqr-neg 43.4%

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

      7. sqrt-unprod 13.4%

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

      8. add-sqr-sqrt 60.5%

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

      9. sub-neg 60.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 82.5%

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

      12. sqr-neg 82.5%

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

      13. sqrt-unprod 82.3%

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

      14. add-sqr-sqrt 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. exp-1-e 82.5%

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

      2. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   if 2.45000000000000011e81 < eps < 1.11999999999999996e94

   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%

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

   5. Taylor expanded in x around 0 96.5%

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

      1. metadata-eval 96.5%

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

      2. distribute-neg-frac 96.5%

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

      3. metadata-eval 96.5%

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

      4. associate-*l/ 96.5%

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

      5. *-commutative 96.5%

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

      6. distribute-lft-neg-in 96.5%

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

      7. cancel-sign-sub-inv 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*l/ 96.5%

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

      10. metadata-eval 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in eps around inf 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*r* 96.5%

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

      3. neg-mul-1 96.5%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. exp-prod 96.5%

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

      13. distribute-lft-in 96.5%

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

      14. *-rgt-identity 96.5%

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

      15. cancel-sign-sub-inv 96.5%

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

      16. neg-mul-1 96.5%

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

      17. *-commutative 96.5%

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

      18. distribute-lft-out-- 96.5%

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

   10. Simplified 96.5%

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

   if 1.11999999999999996e94 < eps

   1. Initial program 100.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.1%

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

      1. metadata-eval 63.1%

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

      2. distribute-neg-frac 63.1%

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

      3. metadata-eval 63.1%

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

      4. associate-*l/ 63.1%

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

      5. *-commutative 63.1%

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

      6. distribute-lft-neg-in 63.1%

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

      7. cancel-sign-sub-inv 63.1%

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

      8. *-commutative 63.1%

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

      9. associate-*l/ 63.1%

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

      10. metadata-eval 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in eps around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 63.1%

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

      3. neg-mul-1 63.1%

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

      4. exp-prod 47.2%

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

      5. +-commutative 47.2%

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

      6. cancel-sign-sub-inv 47.2%

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

      7. distribute-lft-neg-in 47.2%

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

      8. distribute-rgt-neg-in 47.2%

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

      9. metadata-eval 47.2%

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

      10. *-rgt-identity 47.2%

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

      11. +-commutative 47.2%

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

      12. exp-prod 63.1%

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

      13. distribute-lft-in 63.1%

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

      14. *-rgt-identity 63.1%

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

      15. cancel-sign-sub-inv 63.1%

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

      16. neg-mul-1 63.1%

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

      17. *-commutative 63.1%

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

      18. distribute-lft-out-- 63.1%

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

   10. Simplified 63.1%

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

   11. Taylor expanded in x around 0 85.1%

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

       1. distribute-lft-in 85.1%

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

       2. metadata-eval 85.1%

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

       3. neg-mul-1 85.1%

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

       4. sub-neg 85.1%

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

       5. +-commutative 85.1%

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

   13. Simplified 85.1%

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

   14. Taylor expanded in eps around 0 86.6%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{\left(x + \varepsilon \cdot \left(2 \cdot x + \varepsilon \cdot x\right)\right)} + \left(-1 - \varepsilon\right)\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.45 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {e}^{\left(x \cdot \left(1 + \varepsilon\right)\right)} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.12 \cdot 10^{+94}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {e}^{\left(x \cdot \left(1 + \varepsilon\right)\right)} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2900000000.0)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (if (<= eps 2.9e+81)
     (/
      (+ (* (+ 1.0 (/ 1.0 eps)) (pow E (* x (+ 1.0 eps)))) (- 1.0 (/ 1.0 eps)))
      2.0)
     (if (<= eps 3.1e+94)
       (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
       (/
        (+
         2.0
         (* x (+ (- -1.0 eps) (* 0.5 (+ x (* eps (+ (* x 2.0) (* eps x))))))))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 2.9e+81) {
		tmp = (((1.0 + (1.0 / eps)) * pow(((double) M_E), (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (eps <= 3.1e+94) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 2.9e+81) {
		tmp = (((1.0 + (1.0 / eps)) * Math.pow(Math.E, (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (eps <= 3.1e+94) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2900000000.0:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	elif eps <= 2.9e+81:
		tmp = (((1.0 + (1.0 / eps)) * math.pow(math.e, (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0
	elif eps <= 3.1e+94:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2900000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (eps <= 2.9e+81)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * (exp(1) ^ Float64(x * Float64(1.0 + eps)))) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (eps <= 3.1e+94)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x + Float64(eps * Float64(Float64(x * 2.0) + Float64(eps * x)))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2900000000.0)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	elseif (eps <= 2.9e+81)
		tmp = (((1.0 + (1.0 / eps)) * (2.71828182845904523536 ^ (x * (1.0 + eps)))) + (1.0 - (1.0 / eps))) / 2.0;
	elseif (eps <= 3.1e+94)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2900000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.9e+81], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Power[E, N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 3.1e+94], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x + N[(eps * N[(N[(x * 2.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 2.9e9

   1. Initial program 57.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in eps around 0 38.0%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in eps around 0 81.3%

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

   if 2.9e9 < eps < 2.9e81

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 82.4%

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

      1. *-un-lft-identity 82.4%

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

      2. exp-prod 82.5%

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

      3. *-commutative 82.5%

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

      4. add-sqr-sqrt 30.0%

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

      5. sqrt-unprod 43.4%

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

      6. sqr-neg 43.4%

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

      7. sqrt-unprod 13.4%

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

      8. add-sqr-sqrt 60.5%

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

      9. sub-neg 60.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 82.5%

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

      12. sqr-neg 82.5%

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

      13. sqrt-unprod 82.3%

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

      14. add-sqr-sqrt 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. exp-1-e 82.5%

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

      2. +-commutative 82.5%

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

   9. Simplified 82.5%

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

   if 2.9e81 < eps < 3.09999999999999991e94

   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%

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

   5. Taylor expanded in x around 0 96.5%

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

      1. metadata-eval 96.5%

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

      2. distribute-neg-frac 96.5%

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

      3. metadata-eval 96.5%

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

      4. associate-*l/ 96.5%

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

      5. *-commutative 96.5%

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

      6. distribute-lft-neg-in 96.5%

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

      7. cancel-sign-sub-inv 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*l/ 96.5%

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

      10. metadata-eval 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in eps around inf 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*r* 96.5%

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

      3. neg-mul-1 96.5%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. exp-prod 96.5%

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

      13. distribute-lft-in 96.5%

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

      14. *-rgt-identity 96.5%

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

      15. cancel-sign-sub-inv 96.5%

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

      16. neg-mul-1 96.5%

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

      17. *-commutative 96.5%

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

      18. distribute-lft-out-- 96.5%

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

   10. Simplified 96.5%

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

   if 3.09999999999999991e94 < eps

   1. Initial program 100.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.1%

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

      1. metadata-eval 63.1%

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

      2. distribute-neg-frac 63.1%

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

      3. metadata-eval 63.1%

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

      4. associate-*l/ 63.1%

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

      5. *-commutative 63.1%

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

      6. distribute-lft-neg-in 63.1%

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

      7. cancel-sign-sub-inv 63.1%

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

      8. *-commutative 63.1%

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

      9. associate-*l/ 63.1%

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

      10. metadata-eval 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in eps around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 63.1%

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

      3. neg-mul-1 63.1%

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

      4. exp-prod 47.2%

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

      5. +-commutative 47.2%

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

      6. cancel-sign-sub-inv 47.2%

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

      7. distribute-lft-neg-in 47.2%

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

      8. distribute-rgt-neg-in 47.2%

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

      9. metadata-eval 47.2%

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

      10. *-rgt-identity 47.2%

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

      11. +-commutative 47.2%

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

      12. exp-prod 63.1%

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

      13. distribute-lft-in 63.1%

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

      14. *-rgt-identity 63.1%

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

      15. cancel-sign-sub-inv 63.1%

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

      16. neg-mul-1 63.1%

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

      17. *-commutative 63.1%

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

      18. distribute-lft-out-- 63.1%

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

   10. Simplified 63.1%

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

   11. Taylor expanded in x around 0 85.1%

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

       1. distribute-lft-in 85.1%

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

       2. metadata-eval 85.1%

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

       3. neg-mul-1 85.1%

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

       4. sub-neg 85.1%

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

       5. +-commutative 85.1%

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

   13. Simplified 85.1%

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

   14. Taylor expanded in eps around 0 86.6%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{\left(x + \varepsilon \cdot \left(2 \cdot x + \varepsilon \cdot x\right)\right)} + \left(-1 - \varepsilon\right)\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.9 \cdot 10^{+81}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot {e}^{\left(x \cdot \left(1 + \varepsilon\right)\right)} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{+94}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2900000000.0)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (if (<= eps 7e+80)
     (/
      (+ (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0)))) (- 1.0 (/ 1.0 eps)))
      2.0)
     (if (<= eps 3.4e+93)
       (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
       (/
        (+
         2.0
         (* x (+ (- -1.0 eps) (* 0.5 (+ x (* eps (+ (* x 2.0) (* eps x))))))))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 7e+80) {
		tmp = (((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (eps <= 3.4e+93) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 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 (eps <= 2900000000.0d0) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if (eps <= 7d+80) then
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    else if (eps <= 3.4d+93) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((-1.0d0) - eps) + (0.5d0 * (x + (eps * ((x * 2.0d0) + (eps * x)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 7e+80) {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (eps <= 3.4e+93) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2900000000.0:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	elif eps <= 7e+80:
		tmp = (((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) + (1.0 - (1.0 / eps))) / 2.0
	elif eps <= 3.4e+93:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2900000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (eps <= 7e+80)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (eps <= 3.4e+93)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x + Float64(eps * Float64(Float64(x * 2.0) + Float64(eps * x)))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2900000000.0)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	elseif (eps <= 7e+80)
		tmp = (((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (1.0 - (1.0 / eps))) / 2.0;
	elseif (eps <= 3.4e+93)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2900000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 7e+80], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 3.4e+93], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x + N[(eps * N[(N[(x * 2.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 2.9e9

   1. Initial program 57.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in eps around 0 38.0%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in eps around 0 81.3%

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

   if 2.9e9 < eps < 6.99999999999999987e80

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 82.4%

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

   if 6.99999999999999987e80 < eps < 3.4e93

   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%

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

   5. Taylor expanded in x around 0 96.5%

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

      1. metadata-eval 96.5%

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

      2. distribute-neg-frac 96.5%

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

      3. metadata-eval 96.5%

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

      4. associate-*l/ 96.5%

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

      5. *-commutative 96.5%

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

      6. distribute-lft-neg-in 96.5%

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

      7. cancel-sign-sub-inv 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*l/ 96.5%

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

      10. metadata-eval 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in eps around inf 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*r* 96.5%

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

      3. neg-mul-1 96.5%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. exp-prod 96.5%

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

      13. distribute-lft-in 96.5%

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

      14. *-rgt-identity 96.5%

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

      15. cancel-sign-sub-inv 96.5%

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

      16. neg-mul-1 96.5%

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

      17. *-commutative 96.5%

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

      18. distribute-lft-out-- 96.5%

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

   10. Simplified 96.5%

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

   if 3.4e93 < eps

   1. Initial program 100.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.1%

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

      1. metadata-eval 63.1%

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

      2. distribute-neg-frac 63.1%

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

      3. metadata-eval 63.1%

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

      4. associate-*l/ 63.1%

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

      5. *-commutative 63.1%

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

      6. distribute-lft-neg-in 63.1%

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

      7. cancel-sign-sub-inv 63.1%

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

      8. *-commutative 63.1%

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

      9. associate-*l/ 63.1%

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

      10. metadata-eval 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in eps around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 63.1%

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

      3. neg-mul-1 63.1%

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

      4. exp-prod 47.2%

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

      5. +-commutative 47.2%

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

      6. cancel-sign-sub-inv 47.2%

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

      7. distribute-lft-neg-in 47.2%

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

      8. distribute-rgt-neg-in 47.2%

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

      9. metadata-eval 47.2%

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

      10. *-rgt-identity 47.2%

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

      11. +-commutative 47.2%

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

      12. exp-prod 63.1%

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

      13. distribute-lft-in 63.1%

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

      14. *-rgt-identity 63.1%

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

      15. cancel-sign-sub-inv 63.1%

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

      16. neg-mul-1 63.1%

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

      17. *-commutative 63.1%

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

      18. distribute-lft-out-- 63.1%

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

   10. Simplified 63.1%

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

   11. Taylor expanded in x around 0 85.1%

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

       1. distribute-lft-in 85.1%

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

       2. metadata-eval 85.1%

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

       3. neg-mul-1 85.1%

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

       4. sub-neg 85.1%

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

       5. +-commutative 85.1%

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

   13. Simplified 85.1%

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

   14. Taylor expanded in eps around 0 86.6%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{\left(x + \varepsilon \cdot \left(2 \cdot x + \varepsilon \cdot x\right)\right)} + \left(-1 - \varepsilon\right)\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 7 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+169} \lor \neg \left(x \leq 5.5 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3e-285)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (or (<= x 5.8e+169) (not (<= x 5.5e+263)))
     (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3e-285) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if ((x <= 5.8e+169) || !(x <= 5.5e+263)) {
		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 <= 3d-285) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if ((x <= 5.8d+169) .or. (.not. (x <= 5.5d+263))) 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 <= 3e-285) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if ((x <= 5.8e+169) || !(x <= 5.5e+263)) {
		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 <= 3e-285:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif (x <= 5.8e+169) or not (x <= 5.5e+263):
		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 <= 3e-285)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif ((x <= 5.8e+169) || !(x <= 5.5e+263))
		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 <= 3e-285)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif ((x <= 5.8e+169) || ~((x <= 5.5e+263)))
		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, 3e-285], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5.8e+169], N[Not[LessEqual[x, 5.5e+263]], $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 < 3.00000000000000003e-285

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in x around 0 58.2%

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

      1. metadata-eval 58.2%

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

      2. distribute-neg-frac 58.2%

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

      3. metadata-eval 58.2%

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

      4. associate-*l/ 58.2%

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

      5. *-commutative 58.2%

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

      6. distribute-lft-neg-in 58.2%

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

      7. cancel-sign-sub-inv 58.2%

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

      8. *-commutative 58.2%

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

      9. associate-*l/ 58.2%

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

      10. metadata-eval 58.2%

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

   7. Simplified 58.2%

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

   8. Taylor expanded in eps around inf 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*r* 85.2%

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

      3. neg-mul-1 85.2%

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

      4. exp-prod 76.1%

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

      5. +-commutative 76.1%

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

      6. cancel-sign-sub-inv 76.1%

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

      7. distribute-lft-neg-in 76.1%

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

      8. distribute-rgt-neg-in 76.1%

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

      9. metadata-eval 76.1%

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

      10. *-rgt-identity 76.1%

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

      11. +-commutative 76.1%

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

      12. exp-prod 85.2%

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

      13. distribute-lft-in 85.2%

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

      14. *-rgt-identity 85.2%

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

      15. cancel-sign-sub-inv 85.2%

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

      16. neg-mul-1 85.2%

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

      17. *-commutative 85.2%

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

      18. distribute-lft-out-- 85.2%

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

   10. Simplified 85.2%

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

   11. Taylor expanded in eps around inf 85.4%

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

       1. mul-1-neg 85.4%

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

       2. distribute-lft-neg-out 85.4%

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

       3. *-commutative 85.4%

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

   13. Simplified 85.4%

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

   if 3.00000000000000003e-285 < x < 5.8000000000000001e169 or 5.5e263 < x

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in x around 0 41.0%

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

   6. Taylor expanded in eps around inf 72.4%

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

      1. exp-prod 72.5%

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

      2. sub-neg 72.5%

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

      3. mul-1-neg 72.5%

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

      4. exp-prod 72.4%

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

      5. mul-1-neg 72.4%

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

      6. mul-1-neg 72.4%

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

      7. sub-neg 72.4%

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

      8. distribute-rgt-neg-in 72.4%

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

      9. metadata-eval 72.4%

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

      10. associate--r+ 72.4%

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

      11. neg-sub0 72.4%

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

      12. remove-double-neg 72.4%

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

      13. +-commutative 72.4%

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

   8. Simplified 72.4%

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

   if 5.8000000000000001e169 < x < 5.5e263

   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%

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

   5. Taylor expanded in eps around 0 71.0%

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

      1. mul-1-neg 71.0%

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

      2. mul-1-neg 71.0%

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

      3. rec-exp 71.0%

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

      4. sub-neg 71.0%

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

      5. div-sub 71.0%

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

      6. mul-1-neg 71.0%

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

      7. rec-exp 71.0%

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

      8. +-inverses 71.0%

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

   7. Simplified 71.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+169} \lor \neg \left(x \leq 5.5 \cdot 10^{+263}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.32 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2900000000.0)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (if (<= eps 1.32e+81)
     (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
     (if (<= eps 4.4e+93)
       (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
       (/
        (+
         2.0
         (* x (+ (- -1.0 eps) (* 0.5 (+ x (* eps (+ (* x 2.0) (* eps x))))))))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 1.32e+81) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else if (eps <= 4.4e+93) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 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 (eps <= 2900000000.0d0) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if (eps <= 1.32d+81) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else if (eps <= 4.4d+93) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((-1.0d0) - eps) + (0.5d0 * (x + (eps * ((x * 2.0d0) + (eps * x)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2900000000.0) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 1.32e+81) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else if (eps <= 4.4e+93) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2900000000.0:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	elif eps <= 1.32e+81:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	elif eps <= 4.4e+93:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2900000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (eps <= 1.32e+81)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	elseif (eps <= 4.4e+93)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x + Float64(eps * Float64(Float64(x * 2.0) + Float64(eps * x)))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2900000000.0)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	elseif (eps <= 1.32e+81)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	elseif (eps <= 4.4e+93)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2900000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.32e+81], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 4.4e+93], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x + N[(eps * N[(N[(x * 2.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 2.9e9

   1. Initial program 57.3%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in eps around 0 38.0%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in eps around 0 81.3%

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

   if 2.9e9 < eps < 1.31999999999999996e81

   1. Initial program 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 82.4%

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

   6. Taylor expanded in eps around inf 82.4%

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

      1. exp-prod 82.5%

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

      2. sub-neg 82.5%

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

      3. mul-1-neg 82.5%

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

      4. exp-prod 82.4%

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

      5. mul-1-neg 82.4%

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

      6. mul-1-neg 82.4%

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

      7. sub-neg 82.4%

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

      8. distribute-rgt-neg-in 82.4%

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

      9. metadata-eval 82.4%

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

      10. associate--r+ 82.4%

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

      11. neg-sub0 82.4%

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

      12. remove-double-neg 82.4%

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

      13. +-commutative 82.4%

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

   8. Simplified 82.4%

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

   if 1.31999999999999996e81 < eps < 4.40000000000000042e93

   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%

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

   5. Taylor expanded in x around 0 96.5%

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

      1. metadata-eval 96.5%

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

      2. distribute-neg-frac 96.5%

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

      3. metadata-eval 96.5%

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

      4. associate-*l/ 96.5%

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

      5. *-commutative 96.5%

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

      6. distribute-lft-neg-in 96.5%

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

      7. cancel-sign-sub-inv 96.5%

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

      8. *-commutative 96.5%

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

      9. associate-*l/ 96.5%

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

      10. metadata-eval 96.5%

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

   7. Simplified 96.5%

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

   8. Taylor expanded in eps around inf 96.5%

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

      1. *-commutative 96.5%

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

      2. associate-*r* 96.5%

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

      3. neg-mul-1 96.5%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. cancel-sign-sub-inv 100.0%

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

      7. distribute-lft-neg-in 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. exp-prod 96.5%

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

      13. distribute-lft-in 96.5%

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

      14. *-rgt-identity 96.5%

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

      15. cancel-sign-sub-inv 96.5%

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

      16. neg-mul-1 96.5%

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

      17. *-commutative 96.5%

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

      18. distribute-lft-out-- 96.5%

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

   10. Simplified 96.5%

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

   if 4.40000000000000042e93 < eps

   1. Initial program 100.0%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.1%

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

      1. metadata-eval 63.1%

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

      2. distribute-neg-frac 63.1%

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

      3. metadata-eval 63.1%

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

      4. associate-*l/ 63.1%

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

      5. *-commutative 63.1%

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

      6. distribute-lft-neg-in 63.1%

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

      7. cancel-sign-sub-inv 63.1%

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

      8. *-commutative 63.1%

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

      9. associate-*l/ 63.1%

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

      10. metadata-eval 63.1%

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

   7. Simplified 63.1%

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

   8. Taylor expanded in eps around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. associate-*r* 63.1%

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

      3. neg-mul-1 63.1%

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

      4. exp-prod 47.2%

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

      5. +-commutative 47.2%

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

      6. cancel-sign-sub-inv 47.2%

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

      7. distribute-lft-neg-in 47.2%

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

      8. distribute-rgt-neg-in 47.2%

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

      9. metadata-eval 47.2%

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

      10. *-rgt-identity 47.2%

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

      11. +-commutative 47.2%

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

      12. exp-prod 63.1%

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

      13. distribute-lft-in 63.1%

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

      14. *-rgt-identity 63.1%

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

      15. cancel-sign-sub-inv 63.1%

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

      16. neg-mul-1 63.1%

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

      17. *-commutative 63.1%

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

      18. distribute-lft-out-- 63.1%

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

   10. Simplified 63.1%

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

   11. Taylor expanded in x around 0 85.1%

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

       1. distribute-lft-in 85.1%

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

       2. metadata-eval 85.1%

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

       3. neg-mul-1 85.1%

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

       4. sub-neg 85.1%

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

       5. +-commutative 85.1%

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

   13. Simplified 85.1%

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

   14. Taylor expanded in eps around 0 86.6%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{\left(x + \varepsilon \cdot \left(2 \cdot x + \varepsilon \cdot x\right)\right)} + \left(-1 - \varepsilon\right)\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2900000000:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.32 \cdot 10^{+81}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-175}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+265}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3e-175)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 8e+39)
     (/
      (+ 2.0 (* x (+ (- -1.0 eps) (* 0.5 (* x (+ 1.0 (* eps (+ eps 2.0))))))))
      2.0)
     (if (<= x 4.5e+84)
       0.0
       (if (<= x 5.4e+169)
         (/
          (+
           2.0
           (*
            x
            (+ (- -1.0 eps) (* 0.5 (+ x (* eps (+ (* x 2.0) (* eps x))))))))
          2.0)
         (if (<= x 2.6e+265) 0.0 (/ (* eps (+ x (/ 2.0 eps))) 2.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3e-175) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 8e+39) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0;
	} else if (x <= 4.5e+84) {
		tmp = 0.0;
	} else if (x <= 5.4e+169) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	} else if (x <= 2.6e+265) {
		tmp = 0.0;
	} else {
		tmp = (eps * (x + (2.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 3d-175) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 8d+39) then
        tmp = (2.0d0 + (x * (((-1.0d0) - eps) + (0.5d0 * (x * (1.0d0 + (eps * (eps + 2.0d0)))))))) / 2.0d0
    else if (x <= 4.5d+84) then
        tmp = 0.0d0
    else if (x <= 5.4d+169) then
        tmp = (2.0d0 + (x * (((-1.0d0) - eps) + (0.5d0 * (x + (eps * ((x * 2.0d0) + (eps * x)))))))) / 2.0d0
    else if (x <= 2.6d+265) then
        tmp = 0.0d0
    else
        tmp = (eps * (x + (2.0d0 / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3e-175) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 8e+39) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0;
	} else if (x <= 4.5e+84) {
		tmp = 0.0;
	} else if (x <= 5.4e+169) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	} else if (x <= 2.6e+265) {
		tmp = 0.0;
	} else {
		tmp = (eps * (x + (2.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3e-175:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 8e+39:
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0
	elif x <= 4.5e+84:
		tmp = 0.0
	elif x <= 5.4e+169:
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0
	elif x <= 2.6e+265:
		tmp = 0.0
	else:
		tmp = (eps * (x + (2.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3e-175)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 8e+39)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x * Float64(1.0 + Float64(eps * Float64(eps + 2.0)))))))) / 2.0);
	elseif (x <= 4.5e+84)
		tmp = 0.0;
	elseif (x <= 5.4e+169)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x + Float64(eps * Float64(Float64(x * 2.0) + Float64(eps * x)))))))) / 2.0);
	elseif (x <= 2.6e+265)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps * Float64(x + Float64(2.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3e-175)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 8e+39)
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0;
	elseif (x <= 4.5e+84)
		tmp = 0.0;
	elseif (x <= 5.4e+169)
		tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * ((x * 2.0) + (eps * x)))))))) / 2.0;
	elseif (x <= 2.6e+265)
		tmp = 0.0;
	else
		tmp = (eps * (x + (2.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3e-175], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8e+39], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x * N[(1.0 + N[(eps * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.5e+84], 0.0, If[LessEqual[x, 5.4e+169], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x + N[(eps * N[(N[(x * 2.0), $MachinePrecision] + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e+265], 0.0, N[(N[(eps * N[(x + N[(2.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 3e-175

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in x around 0 55.1%

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

      1. metadata-eval 55.1%

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

      2. distribute-neg-frac 55.1%

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

      3. metadata-eval 55.1%

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

      4. associate-*l/ 55.1%

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

      5. *-commutative 55.1%

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

      6. distribute-lft-neg-in 55.1%

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

      7. cancel-sign-sub-inv 55.1%

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

      8. *-commutative 55.1%

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

      9. associate-*l/ 55.1%

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

      10. metadata-eval 55.1%

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

   7. Simplified 55.1%

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

   8. Taylor expanded in eps around inf 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-*r* 86.5%

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

      3. neg-mul-1 86.5%

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

      4. exp-prod 79.0%

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

      5. +-commutative 79.0%

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

      6. cancel-sign-sub-inv 79.0%

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

      7. distribute-lft-neg-in 79.0%

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

      8. distribute-rgt-neg-in 79.0%

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

      9. metadata-eval 79.0%

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

      10. *-rgt-identity 79.0%

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

      11. +-commutative 79.0%

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

      12. exp-prod 86.5%

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

      13. distribute-lft-in 86.5%

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

      14. *-rgt-identity 86.5%

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

      15. cancel-sign-sub-inv 86.5%

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

      16. neg-mul-1 86.5%

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

      17. *-commutative 86.5%

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

      18. distribute-lft-out-- 86.5%

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

   10. Simplified 86.5%

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

   11. Taylor expanded in eps around inf 86.7%

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

       1. mul-1-neg 86.7%

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

       2. distribute-lft-neg-out 86.7%

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

       3. *-commutative 86.7%

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

   13. Simplified 86.7%

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

   if 3e-175 < x < 7.99999999999999952e39

   1. Initial program 52.3%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in x around 0 30.1%

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

      1. metadata-eval 30.1%

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

      2. distribute-neg-frac 30.1%

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

      3. metadata-eval 30.1%

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

      4. associate-*l/ 30.1%

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

      5. *-commutative 30.1%

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

      6. distribute-lft-neg-in 30.1%

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

      7. cancel-sign-sub-inv 30.1%

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

      8. *-commutative 30.1%

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

      9. associate-*l/ 30.1%

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

      10. metadata-eval 30.1%

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

   7. Simplified 30.1%

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

   8. Taylor expanded in eps around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*r* 75.0%

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

      3. neg-mul-1 75.0%

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

      4. exp-prod 70.0%

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

      5. +-commutative 70.0%

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

      6. cancel-sign-sub-inv 70.0%

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

      7. distribute-lft-neg-in 70.0%

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

      8. distribute-rgt-neg-in 70.0%

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

      9. metadata-eval 70.0%

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

      10. *-rgt-identity 70.0%

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

      11. +-commutative 70.0%

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

      12. exp-prod 75.0%

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

      13. distribute-lft-in 75.0%

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

      14. *-rgt-identity 75.0%

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

      15. cancel-sign-sub-inv 75.0%

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

      16. neg-mul-1 75.0%

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

      17. *-commutative 75.0%

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

      18. distribute-lft-out-- 75.0%

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

   10. Simplified 75.0%

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

   11. Taylor expanded in x around 0 84.3%

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

       1. distribute-lft-in 84.3%

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

       2. metadata-eval 84.3%

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

       3. neg-mul-1 84.3%

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

       4. sub-neg 84.3%

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

       5. +-commutative 84.3%

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

   13. Simplified 84.3%

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

   14. Taylor expanded in eps around 0 84.3%

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

   if 7.99999999999999952e39 < x < 4.4999999999999997e84 or 5.39999999999999981e169 < x < 2.6000000000000001e265

   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%

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

   5. Taylor expanded in eps around 0 69.5%

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

      1. mul-1-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. rec-exp 69.5%

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

      4. sub-neg 69.5%

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

      5. div-sub 69.5%

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

      6. mul-1-neg 69.5%

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

      7. rec-exp 69.5%

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

      8. +-inverses 69.5%

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

   7. Simplified 69.5%

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

   if 4.4999999999999997e84 < x < 5.39999999999999981e169

   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%

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

   5. Taylor expanded in x around 0 8.5%

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

      1. metadata-eval 8.5%

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

      2. distribute-neg-frac 8.5%

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

      3. metadata-eval 8.5%

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

      4. associate-*l/ 8.5%

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

      5. *-commutative 8.5%

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

      6. distribute-lft-neg-in 8.5%

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

      7. cancel-sign-sub-inv 8.5%

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

      8. *-commutative 8.5%

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

      9. associate-*l/ 8.5%

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

      10. metadata-eval 8.5%

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

   7. Simplified 8.5%

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

   8. Taylor expanded in eps around inf 8.8%

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

      1. *-commutative 8.8%

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

      2. associate-*r* 8.8%

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

      3. neg-mul-1 8.8%

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

      4. exp-prod 8.8%

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

      5. +-commutative 8.8%

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

      6. cancel-sign-sub-inv 8.8%

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

      7. distribute-lft-neg-in 8.8%

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

      8. distribute-rgt-neg-in 8.8%

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

      9. metadata-eval 8.8%

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

      10. *-rgt-identity 8.8%

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

      11. +-commutative 8.8%

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

      12. exp-prod 8.8%

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

      13. distribute-lft-in 8.8%

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

      14. *-rgt-identity 8.8%

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

      15. cancel-sign-sub-inv 8.8%

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

      16. neg-mul-1 8.8%

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

      17. *-commutative 8.8%

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

      18. distribute-lft-out-- 8.8%

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

   10. Simplified 8.8%

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

   11. Taylor expanded in x around 0 65.4%

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

       1. distribute-lft-in 65.4%

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

       2. metadata-eval 65.4%

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

       3. neg-mul-1 65.4%

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

       4. sub-neg 65.4%

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

       5. +-commutative 65.4%

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

   13. Simplified 65.4%

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

   14. Taylor expanded in eps around 0 65.4%

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

   if 2.6000000000000001e265 < 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%

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

   5. Taylor expanded in x around 0 37.9%

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

      1. metadata-eval 37.9%

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

      2. distribute-neg-frac 37.9%

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

      3. metadata-eval 37.9%

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

      4. associate-*l/ 37.9%

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

      5. *-commutative 37.9%

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

      6. distribute-lft-neg-in 37.9%

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

      7. cancel-sign-sub-inv 37.9%

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

      8. *-commutative 37.9%

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

      9. associate-*l/ 37.9%

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

      10. metadata-eval 37.9%

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

   7. Simplified 37.9%

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

   8. Taylor expanded in x around 0 36.9%

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

   9. Taylor expanded in eps around -inf 37.3%

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

       1. mul-1-neg 37.3%

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

       2. *-commutative 37.3%

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

       3. distribute-rgt-neg-in 37.3%

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

       4. sub-neg 37.3%

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

       5. associate-*r/ 37.3%

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

       6. metadata-eval 37.3%

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

       7. distribute-neg-frac 37.3%

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

       8. metadata-eval 37.3%

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

   11. Simplified 37.3%

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

       1. pow1 37.3%

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

       2. *-commutative 37.3%

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

       3. add-sqr-sqrt 36.9%

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

       4. sqrt-unprod 91.1%

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

       5. sqr-neg 91.1%

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

       6. sqrt-unprod 27.7%

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

       7. add-sqr-sqrt 28.2%

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

       8. frac-2neg 28.2%

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

       9. metadata-eval 28.2%

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

       10. add-sqr-sqrt 0.5%

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

       11. sqrt-unprod 27.9%

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

       12. sqr-neg 27.9%

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

       13. sqrt-unprod 27.7%

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

       14. add-sqr-sqrt 28.2%

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

   13. Applied egg-rr 28.2%

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

       1. unpow1 28.2%

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

   15. Simplified 28.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-175}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+39}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+169}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot 2 + \varepsilon \cdot x\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+265}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - \varepsilon \cdot x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 440:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\frac{2 + \left(1 - \varepsilon\right) \cdot \left(x \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.4e-29)
   (/ (/ (+ x (* eps (- 2.0 (* eps x)))) eps) 2.0)
   (if (<= x 440.0)
     1.0
     (if (<= x 3.05e+264)
       0.0
       (if (<= x 2e+290)
         (/ (+ 2.0 (* (- 1.0 eps) (* x (+ -1.0 (/ -1.0 eps))))) 2.0)
         0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.4e-29) {
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0;
	} else if (x <= 440.0) {
		tmp = 1.0;
	} else if (x <= 3.05e+264) {
		tmp = 0.0;
	} else if (x <= 2e+290) {
		tmp = (2.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-7.4d-29)) then
        tmp = ((x + (eps * (2.0d0 - (eps * x)))) / eps) / 2.0d0
    else if (x <= 440.0d0) then
        tmp = 1.0d0
    else if (x <= 3.05d+264) then
        tmp = 0.0d0
    else if (x <= 2d+290) then
        tmp = (2.0d0 + ((1.0d0 - eps) * (x * ((-1.0d0) + ((-1.0d0) / eps))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7.4e-29) {
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0;
	} else if (x <= 440.0) {
		tmp = 1.0;
	} else if (x <= 3.05e+264) {
		tmp = 0.0;
	} else if (x <= 2e+290) {
		tmp = (2.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7.4e-29:
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0
	elif x <= 440.0:
		tmp = 1.0
	elif x <= 3.05e+264:
		tmp = 0.0
	elif x <= 2e+290:
		tmp = (2.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.4e-29)
		tmp = Float64(Float64(Float64(x + Float64(eps * Float64(2.0 - Float64(eps * x)))) / eps) / 2.0);
	elseif (x <= 440.0)
		tmp = 1.0;
	elseif (x <= 3.05e+264)
		tmp = 0.0;
	elseif (x <= 2e+290)
		tmp = Float64(Float64(2.0 + Float64(Float64(1.0 - eps) * Float64(x * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7.4e-29)
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0;
	elseif (x <= 440.0)
		tmp = 1.0;
	elseif (x <= 3.05e+264)
		tmp = 0.0;
	elseif (x <= 2e+290)
		tmp = (2.0 + ((1.0 - eps) * (x * (-1.0 + (-1.0 / eps))))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.4e-29], N[(N[(N[(x + N[(eps * N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 440.0], 1.0, If[LessEqual[x, 3.05e+264], 0.0, If[LessEqual[x, 2e+290], N[(N[(2.0 + N[(N[(1.0 - eps), $MachinePrecision] * N[(x * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.3999999999999995e-29

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. metadata-eval 67.7%

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

      2. distribute-neg-frac 67.7%

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

      3. metadata-eval 67.7%

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

      4. associate-*l/ 67.7%

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

      5. *-commutative 67.7%

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

      6. distribute-lft-neg-in 67.7%

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

      7. cancel-sign-sub-inv 67.7%

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

      8. *-commutative 67.7%

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

      9. associate-*l/ 67.7%

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

      10. metadata-eval 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in x around 0 46.7%

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

   9. Taylor expanded in eps around 0 58.0%

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

       1. +-commutative 58.0%

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

       2. mul-1-neg 58.0%

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

       3. distribute-lft-neg-out 58.0%

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

       4. *-commutative 58.0%

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

   11. Simplified 58.0%

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

   if -7.3999999999999995e-29 < x < 440

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 79.5%

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

   if 440 < x < 3.05000000000000014e264 or 2.00000000000000012e290 < 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%

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

   5. Taylor expanded in eps around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

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

      3. rec-exp 55.1%

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

      4. sub-neg 55.1%

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

      5. div-sub 55.1%

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

      6. mul-1-neg 55.1%

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

      7. rec-exp 55.1%

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

      8. +-inverses 55.1%

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

   7. Simplified 55.1%

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

   if 3.05000000000000014e264 < x < 2.00000000000000012e290

   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%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in x around 0 50.0%

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

      1. mul-1-neg 50.0%

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

      2. distribute-rgt-neg-in 50.0%

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

      3. mul-1-neg 50.0%

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

      4. associate-*r* 50.0%

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

      5. associate-*r* 50.0%

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

      6. distribute-lft-in 50.0%

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

      7. metadata-eval 50.0%

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

      8. neg-mul-1 50.0%

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

      9. distribute-neg-frac 50.0%

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

      10. metadata-eval 50.0%

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

   8. Simplified 50.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - \varepsilon \cdot x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 440:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\frac{2 + \left(1 - \varepsilon\right) \cdot \left(x \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - \varepsilon \cdot x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+269}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+285}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7.6e-29)
   (/ (/ (+ x (* eps (- 2.0 (* eps x)))) eps) 2.0)
   (if (<= x 490.0)
     1.0
     (if (<= x 1e+269)
       0.0
       (if (<= x 9e+285)
         (/ (+ 2.0 (* x (+ (- -1.0 eps) (* x 0.5)))) 2.0)
         0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7.6e-29) {
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0;
	} else if (x <= 490.0) {
		tmp = 1.0;
	} else if (x <= 1e+269) {
		tmp = 0.0;
	} else if (x <= 9e+285) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-7.6d-29)) then
        tmp = ((x + (eps * (2.0d0 - (eps * x)))) / eps) / 2.0d0
    else if (x <= 490.0d0) then
        tmp = 1.0d0
    else if (x <= 1d+269) then
        tmp = 0.0d0
    else if (x <= 9d+285) then
        tmp = (2.0d0 + (x * (((-1.0d0) - eps) + (x * 0.5d0)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7.6e-29) {
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0;
	} else if (x <= 490.0) {
		tmp = 1.0;
	} else if (x <= 1e+269) {
		tmp = 0.0;
	} else if (x <= 9e+285) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7.6e-29:
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0
	elif x <= 490.0:
		tmp = 1.0
	elif x <= 1e+269:
		tmp = 0.0
	elif x <= 9e+285:
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7.6e-29)
		tmp = Float64(Float64(Float64(x + Float64(eps * Float64(2.0 - Float64(eps * x)))) / eps) / 2.0);
	elseif (x <= 490.0)
		tmp = 1.0;
	elseif (x <= 1e+269)
		tmp = 0.0;
	elseif (x <= 9e+285)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(x * 0.5)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7.6e-29)
		tmp = ((x + (eps * (2.0 - (eps * x)))) / eps) / 2.0;
	elseif (x <= 490.0)
		tmp = 1.0;
	elseif (x <= 1e+269)
		tmp = 0.0;
	elseif (x <= 9e+285)
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7.6e-29], N[(N[(N[(x + N[(eps * N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 490.0], 1.0, If[LessEqual[x, 1e+269], 0.0, If[LessEqual[x, 9e+285], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.59999999999999951e-29

   1. Initial program 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around 0 67.7%

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

      1. metadata-eval 67.7%

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

      2. distribute-neg-frac 67.7%

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

      3. metadata-eval 67.7%

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

      4. associate-*l/ 67.7%

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

      5. *-commutative 67.7%

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

      6. distribute-lft-neg-in 67.7%

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

      7. cancel-sign-sub-inv 67.7%

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

      8. *-commutative 67.7%

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

      9. associate-*l/ 67.7%

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

      10. metadata-eval 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in x around 0 46.7%

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

   9. Taylor expanded in eps around 0 58.0%

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

       1. +-commutative 58.0%

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

       2. mul-1-neg 58.0%

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

       3. distribute-lft-neg-out 58.0%

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

       4. *-commutative 58.0%

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

   11. Simplified 58.0%

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

   if -7.59999999999999951e-29 < x < 490

   1. Initial program 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 79.5%

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

   if 490 < x < 1e269 or 9.0000000000000001e285 < 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%

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

   5. Taylor expanded in eps around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 55.1%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 55.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 55.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 55.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 55.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1e269 < x < 9.0000000000000001e285

   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%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.6%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 51.6%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 51.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 51.6%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 51.6%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 51.6%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 51.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 51.6%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 51.6%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 51.6%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 51.6%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 51.6%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in eps around inf 51.6%

      \[\leadsto \frac{\color{blue}{1 - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 51.6%

         \[\leadsto \frac{1 - \color{blue}{e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot -1}}{2} \]

      2. associate-*r* 51.6%

         \[\leadsto \frac{1 - e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}} \cdot -1}{2} \]

      3. neg-mul-1 51.6%

         \[\leadsto \frac{1 - e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)} \cdot -1}{2} \]

      4. exp-prod 51.6%

         \[\leadsto \frac{1 - \color{blue}{{\left(e^{-x}\right)}^{\left(1 + \varepsilon\right)}} \cdot -1}{2} \]

      5. +-commutative 51.6%

         \[\leadsto \frac{1 - {\left(e^{-x}\right)}^{\color{blue}{\left(\varepsilon + 1\right)}} \cdot -1}{2} \]

      6. cancel-sign-sub-inv 51.6%

         \[\leadsto \frac{\color{blue}{1 + \left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}\right) \cdot -1}}{2} \]

      7. distribute-lft-neg-in 51.6%

         \[\leadsto \frac{1 + \color{blue}{\left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot -1\right)}}{2} \]

      8. distribute-rgt-neg-in 51.6%

         \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(--1\right)}}{2} \]

      9. metadata-eval 51.6%

         \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \color{blue}{1}}{2} \]

      10. *-rgt-identity 51.6%

          \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}}{2} \]

      11. +-commutative 51.6%

          \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\color{blue}{\left(1 + \varepsilon\right)}}}{2} \]

      12. exp-prod 51.6%

          \[\leadsto \frac{1 + \color{blue}{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}}{2} \]

      13. distribute-lft-in 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot 1 + \left(-x\right) \cdot \varepsilon}}}{2} \]

      14. *-rgt-identity 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right)} + \left(-x\right) \cdot \varepsilon}}{2} \]

      15. cancel-sign-sub-inv 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) - x \cdot \varepsilon}}}{2} \]

      16. neg-mul-1 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{-1 \cdot x} - x \cdot \varepsilon}}{2} \]

      17. *-commutative 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot -1} - x \cdot \varepsilon}}{2} \]

      18. distribute-lft-out-- 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   10. Simplified 51.6%

       \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. distribute-lft-in 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       3. neg-mul-1 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       4. sub-neg 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 - \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       5. +-commutative 100.0%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

   14. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{x} + \left(-1 - \varepsilon\right)\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{x + \varepsilon \cdot \left(2 - \varepsilon \cdot x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+269}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+285}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 280 \lor \neg \left(x \leq 1.5 \cdot 10^{+264}\right) \land x \leq 3.6 \cdot 10^{+289}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 280.0) (and (not (<= x 1.5e+264)) (<= x 3.6e+289)))
   (/ (+ 2.0 (* x (+ (- -1.0 eps) (* x 0.5)))) 2.0)
   0.0))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 280.0) || (!(x <= 1.5e+264) && (x <= 3.6e+289))) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 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 <= 280.0d0) .or. (.not. (x <= 1.5d+264)) .and. (x <= 3.6d+289)) then
        tmp = (2.0d0 + (x * (((-1.0d0) - eps) + (x * 0.5d0)))) / 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 <= 280.0) || (!(x <= 1.5e+264) && (x <= 3.6e+289))) {
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= 280.0) or (not (x <= 1.5e+264) and (x <= 3.6e+289)):
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= 280.0) || (!(x <= 1.5e+264) && (x <= 3.6e+289)))
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(x * 0.5)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 280.0) || (~((x <= 1.5e+264)) && (x <= 3.6e+289)))
		tmp = (2.0 + (x * ((-1.0 - eps) + (x * 0.5)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, 280.0], And[N[Not[LessEqual[x, 1.5e+264]], $MachinePrecision], LessEqual[x, 3.6e+289]]], N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 280 or 1.5e264 < x < 3.60000000000000014e289

   1. Initial program 63.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 48.4%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 48.4%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 48.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 48.4%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 48.4%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 48.4%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 48.4%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 48.4%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 48.4%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 48.4%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 48.4%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 48.4%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in eps around inf 83.5%

      \[\leadsto \frac{\color{blue}{1 - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto \frac{1 - \color{blue}{e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot -1}}{2} \]

      2. associate-*r* 83.5%

         \[\leadsto \frac{1 - e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}} \cdot -1}{2} \]

      3. neg-mul-1 83.5%

         \[\leadsto \frac{1 - e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)} \cdot -1}{2} \]

      4. exp-prod 76.6%

         \[\leadsto \frac{1 - \color{blue}{{\left(e^{-x}\right)}^{\left(1 + \varepsilon\right)}} \cdot -1}{2} \]

      5. +-commutative 76.6%

         \[\leadsto \frac{1 - {\left(e^{-x}\right)}^{\color{blue}{\left(\varepsilon + 1\right)}} \cdot -1}{2} \]

      6. cancel-sign-sub-inv 76.6%

         \[\leadsto \frac{\color{blue}{1 + \left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}\right) \cdot -1}}{2} \]

      7. distribute-lft-neg-in 76.6%

         \[\leadsto \frac{1 + \color{blue}{\left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot -1\right)}}{2} \]

      8. distribute-rgt-neg-in 76.6%

         \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(--1\right)}}{2} \]

      9. metadata-eval 76.6%

         \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \color{blue}{1}}{2} \]

      10. *-rgt-identity 76.6%

          \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}}{2} \]

      11. +-commutative 76.6%

          \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\color{blue}{\left(1 + \varepsilon\right)}}}{2} \]

      12. exp-prod 83.5%

          \[\leadsto \frac{1 + \color{blue}{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}}{2} \]

      13. distribute-lft-in 83.5%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot 1 + \left(-x\right) \cdot \varepsilon}}}{2} \]

      14. *-rgt-identity 83.5%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right)} + \left(-x\right) \cdot \varepsilon}}{2} \]

      15. cancel-sign-sub-inv 83.5%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) - x \cdot \varepsilon}}}{2} \]

      16. neg-mul-1 83.5%

          \[\leadsto \frac{1 + e^{\color{blue}{-1 \cdot x} - x \cdot \varepsilon}}{2} \]

      17. *-commutative 83.5%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot -1} - x \cdot \varepsilon}}{2} \]

      18. distribute-lft-out-- 83.5%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   10. Simplified 83.5%

       \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   11. Taylor expanded in x around 0 86.2%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. distribute-lft-in 86.2%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       2. metadata-eval 86.2%

          \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       3. neg-mul-1 86.2%

          \[\leadsto \frac{2 + x \cdot \left(\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       4. sub-neg 86.2%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 - \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       5. +-commutative 86.2%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

   13. Simplified 86.2%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

   14. Taylor expanded in eps around 0 76.0%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{x} + \left(-1 - \varepsilon\right)\right)}{2} \]

   if 280 < x < 1.5e264 or 3.60000000000000014e289 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 55.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

      2. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 55.1%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 55.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 55.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 55.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 55.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 280 \lor \neg \left(x \leq 1.5 \cdot 10^{+264}\right) \land x \leq 3.6 \cdot 10^{+289}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 260:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 260.0)
   (/ (- 2.0 (* eps x)) 2.0)
   (if (<= x 5e+263)
     0.0
     (if (<= x 5e+288) (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 260.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 5e+263) {
		tmp = 0.0;
	} else if (x <= 5e+288) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= 260.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 5d+263) then
        tmp = 0.0d0
    else if (x <= 5d+288) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= 260.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 5e+263) {
		tmp = 0.0;
	} else if (x <= 5e+288) {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 260.0:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 5e+263:
		tmp = 0.0
	elif x <= 5e+288:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 260.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 5e+263)
		tmp = 0.0;
	elseif (x <= 5e+288)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 260.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 5e+263)
		tmp = 0.0;
	elseif (x <= 5e+288)
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 260.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+263], 0.0, If[LessEqual[x, 5e+288], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 260

   1. Initial program 62.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 48.3%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 48.3%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 48.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 48.3%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 48.3%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 48.3%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 48.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 48.3%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 48.3%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 48.3%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 48.3%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 48.3%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in x around 0 54.3%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 73.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(-1 \cdot x + 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 73.4%

          \[\leadsto \frac{\varepsilon \cdot \left(\color{blue}{\left(-x\right)} + 2 \cdot \frac{1}{\varepsilon}\right)}{2} \]

       2. +-commutative 73.4%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(2 \cdot \frac{1}{\varepsilon} + \left(-x\right)\right)}}{2} \]

       3. distribute-lft-in 73.4%

          \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(2 \cdot \frac{1}{\varepsilon}\right) + \varepsilon \cdot \left(-x\right)}}{2} \]

       4. *-commutative 73.4%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\frac{1}{\varepsilon} \cdot 2\right)} + \varepsilon \cdot \left(-x\right)}{2} \]

       5. associate-*r* 73.4%

          \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \frac{1}{\varepsilon}\right) \cdot 2} + \varepsilon \cdot \left(-x\right)}{2} \]

       6. rgt-mult-inverse 73.5%

          \[\leadsto \frac{\color{blue}{1} \cdot 2 + \varepsilon \cdot \left(-x\right)}{2} \]

       7. metadata-eval 73.5%

          \[\leadsto \frac{\color{blue}{2} + \varepsilon \cdot \left(-x\right)}{2} \]

       8. distribute-rgt-neg-in 73.5%

          \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon \cdot x\right)}}{2} \]

       9. unsub-neg 73.5%

          \[\leadsto \frac{\color{blue}{2 - \varepsilon \cdot x}}{2} \]

       10. *-commutative 73.5%

           \[\leadsto \frac{2 - \color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 73.5%

       \[\leadsto \frac{\color{blue}{2 - x \cdot \varepsilon}}{2} \]

   if 260 < x < 5.00000000000000022e263 or 5.0000000000000003e288 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 55.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

      2. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 55.1%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 55.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 55.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 55.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 55.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 5.00000000000000022e263 < x < 5.0000000000000003e288

   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%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.6%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 51.6%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 51.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 51.6%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 51.6%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 51.6%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 51.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 51.6%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 51.6%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 51.6%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 51.6%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 51.6%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in eps around inf 51.6%

      \[\leadsto \frac{\color{blue}{1 - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 51.6%

         \[\leadsto \frac{1 - \color{blue}{e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot -1}}{2} \]

      2. associate-*r* 51.6%

         \[\leadsto \frac{1 - e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}} \cdot -1}{2} \]

      3. neg-mul-1 51.6%

         \[\leadsto \frac{1 - e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)} \cdot -1}{2} \]

      4. exp-prod 51.6%

         \[\leadsto \frac{1 - \color{blue}{{\left(e^{-x}\right)}^{\left(1 + \varepsilon\right)}} \cdot -1}{2} \]

      5. +-commutative 51.6%

         \[\leadsto \frac{1 - {\left(e^{-x}\right)}^{\color{blue}{\left(\varepsilon + 1\right)}} \cdot -1}{2} \]

      6. cancel-sign-sub-inv 51.6%

         \[\leadsto \frac{\color{blue}{1 + \left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}\right) \cdot -1}}{2} \]

      7. distribute-lft-neg-in 51.6%

         \[\leadsto \frac{1 + \color{blue}{\left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot -1\right)}}{2} \]

      8. distribute-rgt-neg-in 51.6%

         \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(--1\right)}}{2} \]

      9. metadata-eval 51.6%

         \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \color{blue}{1}}{2} \]

      10. *-rgt-identity 51.6%

          \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}}{2} \]

      11. +-commutative 51.6%

          \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\color{blue}{\left(1 + \varepsilon\right)}}}{2} \]

      12. exp-prod 51.6%

          \[\leadsto \frac{1 + \color{blue}{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}}{2} \]

      13. distribute-lft-in 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot 1 + \left(-x\right) \cdot \varepsilon}}}{2} \]

      14. *-rgt-identity 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right)} + \left(-x\right) \cdot \varepsilon}}{2} \]

      15. cancel-sign-sub-inv 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) - x \cdot \varepsilon}}}{2} \]

      16. neg-mul-1 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{-1 \cdot x} - x \cdot \varepsilon}}{2} \]

      17. *-commutative 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot -1} - x \cdot \varepsilon}}{2} \]

      18. distribute-lft-out-- 51.6%

          \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   10. Simplified 51.6%

       \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}}{2} \]
   12. Step-by-step derivation

       1. distribute-lft-in 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       2. metadata-eval 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       3. neg-mul-1 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       4. sub-neg 100.0%

          \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 - \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

       5. +-commutative 100.0%

          \[\leadsto \frac{2 + x \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

   13. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

   14. Taylor expanded in eps around 0 100.0%

       \[\leadsto \frac{2 + \color{blue}{x \cdot \left(0.5 \cdot x - 1\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 260:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+263}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+288}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+288}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 235.0)
   (/ (- 2.0 (* eps x)) 2.0)
   (if (<= x 2.7e+264)
     0.0
     (if (<= x 1.02e+288) (/ (* eps (+ x (/ 2.0 eps))) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 235.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.7e+264) {
		tmp = 0.0;
	} else if (x <= 1.02e+288) {
		tmp = (eps * (x + (2.0 / eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 235.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 2.7d+264) then
        tmp = 0.0d0
    else if (x <= 1.02d+288) then
        tmp = (eps * (x + (2.0d0 / eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 235.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.7e+264) {
		tmp = 0.0;
	} else if (x <= 1.02e+288) {
		tmp = (eps * (x + (2.0 / eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 235.0:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 2.7e+264:
		tmp = 0.0
	elif x <= 1.02e+288:
		tmp = (eps * (x + (2.0 / eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 235.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 2.7e+264)
		tmp = 0.0;
	elseif (x <= 1.02e+288)
		tmp = Float64(Float64(eps * Float64(x + Float64(2.0 / eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 235.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 2.7e+264)
		tmp = 0.0;
	elseif (x <= 1.02e+288)
		tmp = (eps * (x + (2.0 / eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 235.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7e+264], 0.0, If[LessEqual[x, 1.02e+288], N[(N[(eps * N[(x + N[(2.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 235

   1. Initial program 62.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 48.3%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 48.3%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 48.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 48.3%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 48.3%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 48.3%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 48.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 48.3%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 48.3%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 48.3%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 48.3%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 48.3%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in x around 0 54.3%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 73.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(-1 \cdot x + 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 73.4%

          \[\leadsto \frac{\varepsilon \cdot \left(\color{blue}{\left(-x\right)} + 2 \cdot \frac{1}{\varepsilon}\right)}{2} \]

       2. +-commutative 73.4%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(2 \cdot \frac{1}{\varepsilon} + \left(-x\right)\right)}}{2} \]

       3. distribute-lft-in 73.4%

          \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(2 \cdot \frac{1}{\varepsilon}\right) + \varepsilon \cdot \left(-x\right)}}{2} \]

       4. *-commutative 73.4%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\frac{1}{\varepsilon} \cdot 2\right)} + \varepsilon \cdot \left(-x\right)}{2} \]

       5. associate-*r* 73.4%

          \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \frac{1}{\varepsilon}\right) \cdot 2} + \varepsilon \cdot \left(-x\right)}{2} \]

       6. rgt-mult-inverse 73.5%

          \[\leadsto \frac{\color{blue}{1} \cdot 2 + \varepsilon \cdot \left(-x\right)}{2} \]

       7. metadata-eval 73.5%

          \[\leadsto \frac{\color{blue}{2} + \varepsilon \cdot \left(-x\right)}{2} \]

       8. distribute-rgt-neg-in 73.5%

          \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon \cdot x\right)}}{2} \]

       9. unsub-neg 73.5%

          \[\leadsto \frac{\color{blue}{2 - \varepsilon \cdot x}}{2} \]

       10. *-commutative 73.5%

           \[\leadsto \frac{2 - \color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 73.5%

       \[\leadsto \frac{\color{blue}{2 - x \cdot \varepsilon}}{2} \]

   if 235 < x < 2.7000000000000002e264 or 1.02000000000000003e288 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 55.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

      2. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 55.1%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 55.1%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 55.1%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 55.1%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 55.1%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 55.1%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 55.1%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 2.7000000000000002e264 < x < 1.02000000000000003e288

   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%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.6%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 51.6%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 51.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 51.6%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 51.6%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 51.6%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 51.6%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 51.6%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 51.6%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 51.6%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 51.6%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 51.6%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in x around 0 50.0%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2} \]

   9. Taylor expanded in eps around -inf 50.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\varepsilon \cdot \left(x - 2 \cdot \frac{1}{\varepsilon}\right)\right)}}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 50.0%

          \[\leadsto \frac{\color{blue}{-\varepsilon \cdot \left(x - 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]

       2. *-commutative 50.0%

          \[\leadsto \frac{-\color{blue}{\left(x - 2 \cdot \frac{1}{\varepsilon}\right) \cdot \varepsilon}}{2} \]

       3. distribute-rgt-neg-in 50.0%

          \[\leadsto \frac{\color{blue}{\left(x - 2 \cdot \frac{1}{\varepsilon}\right) \cdot \left(-\varepsilon\right)}}{2} \]

       4. sub-neg 50.0%

          \[\leadsto \frac{\color{blue}{\left(x + \left(-2 \cdot \frac{1}{\varepsilon}\right)\right)} \cdot \left(-\varepsilon\right)}{2} \]

       5. associate-*r/ 50.0%

          \[\leadsto \frac{\left(x + \left(-\color{blue}{\frac{2 \cdot 1}{\varepsilon}}\right)\right) \cdot \left(-\varepsilon\right)}{2} \]

       6. metadata-eval 50.0%

          \[\leadsto \frac{\left(x + \left(-\frac{\color{blue}{2}}{\varepsilon}\right)\right) \cdot \left(-\varepsilon\right)}{2} \]

       7. distribute-neg-frac 50.0%

          \[\leadsto \frac{\left(x + \color{blue}{\frac{-2}{\varepsilon}}\right) \cdot \left(-\varepsilon\right)}{2} \]

       8. metadata-eval 50.0%

          \[\leadsto \frac{\left(x + \frac{\color{blue}{-2}}{\varepsilon}\right) \cdot \left(-\varepsilon\right)}{2} \]

   11. Simplified 50.0%

       \[\leadsto \frac{\color{blue}{\left(x + \frac{-2}{\varepsilon}\right) \cdot \left(-\varepsilon\right)}}{2} \]
   12. Step-by-step derivation

       1. pow1 50.0%

          \[\leadsto \frac{\color{blue}{{\left(\left(x + \frac{-2}{\varepsilon}\right) \cdot \left(-\varepsilon\right)\right)}^{1}}}{2} \]

       2. *-commutative 50.0%

          \[\leadsto \frac{{\color{blue}{\left(\left(-\varepsilon\right) \cdot \left(x + \frac{-2}{\varepsilon}\right)\right)}}^{1}}{2} \]

       3. add-sqr-sqrt 50.0%

          \[\leadsto \frac{{\left(\color{blue}{\left(\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}\right)} \cdot \left(x + \frac{-2}{\varepsilon}\right)\right)}^{1}}{2} \]

       4. sqrt-unprod 100.0%

          \[\leadsto \frac{{\left(\color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}} \cdot \left(x + \frac{-2}{\varepsilon}\right)\right)}^{1}}{2} \]

       5. sqr-neg 100.0%

          \[\leadsto \frac{{\left(\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}} \cdot \left(x + \frac{-2}{\varepsilon}\right)\right)}^{1}}{2} \]

       6. sqrt-unprod 50.0%

          \[\leadsto \frac{{\left(\color{blue}{\left(\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}\right)} \cdot \left(x + \frac{-2}{\varepsilon}\right)\right)}^{1}}{2} \]

       7. add-sqr-sqrt 50.0%

          \[\leadsto \frac{{\left(\color{blue}{\varepsilon} \cdot \left(x + \frac{-2}{\varepsilon}\right)\right)}^{1}}{2} \]

       8. frac-2neg 50.0%

          \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \color{blue}{\frac{--2}{-\varepsilon}}\right)\right)}^{1}}{2} \]

       9. metadata-eval 50.0%

          \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \frac{\color{blue}{2}}{-\varepsilon}\right)\right)}^{1}}{2} \]

       10. add-sqr-sqrt 0.0%

           \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \frac{2}{\color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}\right)\right)}^{1}}{2} \]

       11. sqrt-unprod 50.0%

           \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \frac{2}{\color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}}\right)\right)}^{1}}{2} \]

       12. sqr-neg 50.0%

           \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \frac{2}{\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}}\right)\right)}^{1}}{2} \]

       13. sqrt-unprod 50.0%

           \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \frac{2}{\color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}\right)\right)}^{1}}{2} \]

       14. add-sqr-sqrt 50.0%

           \[\leadsto \frac{{\left(\varepsilon \cdot \left(x + \frac{2}{\color{blue}{\varepsilon}}\right)\right)}^{1}}{2} \]

   13. Applied egg-rr 50.0%

       \[\leadsto \frac{\color{blue}{{\left(\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)\right)}^{1}}}{2} \]
   14. Step-by-step derivation

       1. unpow1 50.0%

          \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}}{2} \]

   15. Simplified 50.0%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{+288}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{2}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 73.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (+ 2.0 (* x (+ (- -1.0 eps) (* 0.5 (+ x (* eps (* x (+ eps 2.0))))))))
  2.0))

C

double code(double x, double eps) {
	return (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * (x * (eps + 2.0)))))))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 + (x * (((-1.0d0) - eps) + (0.5d0 * (x + (eps * (x * (eps + 2.0d0)))))))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * (x * (eps + 2.0)))))))) / 2.0;
}

Python

def code(x, eps):
	return (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * (x * (eps + 2.0)))))))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x + Float64(eps * Float64(x * Float64(eps + 2.0)))))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x + (eps * (x * (eps + 2.0)))))))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x + N[(eps * N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 72.0%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 41.2%

   \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
6. Step-by-step derivation

   1. metadata-eval 41.2%

      \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   2. distribute-neg-frac 41.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   3. metadata-eval 41.2%

      \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   4. associate-*l/ 41.2%

      \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. *-commutative 41.2%

      \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. distribute-lft-neg-in 41.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. cancel-sign-sub-inv 41.2%

      \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. *-commutative 41.2%

      \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   9. associate-*l/ 41.2%

      \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   10. metadata-eval 41.2%

       \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

7. Simplified 41.2%

   \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

8. Taylor expanded in eps around inf 68.6%

   \[\leadsto \frac{\color{blue}{1 - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
9. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \frac{1 - \color{blue}{e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot -1}}{2} \]

   2. associate-*r* 68.6%

      \[\leadsto \frac{1 - e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}} \cdot -1}{2} \]

   3. neg-mul-1 68.6%

      \[\leadsto \frac{1 - e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)} \cdot -1}{2} \]

   4. exp-prod 63.3%

      \[\leadsto \frac{1 - \color{blue}{{\left(e^{-x}\right)}^{\left(1 + \varepsilon\right)}} \cdot -1}{2} \]

   5. +-commutative 63.3%

      \[\leadsto \frac{1 - {\left(e^{-x}\right)}^{\color{blue}{\left(\varepsilon + 1\right)}} \cdot -1}{2} \]

   6. cancel-sign-sub-inv 63.3%

      \[\leadsto \frac{\color{blue}{1 + \left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}\right) \cdot -1}}{2} \]

   7. distribute-lft-neg-in 63.3%

      \[\leadsto \frac{1 + \color{blue}{\left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot -1\right)}}{2} \]

   8. distribute-rgt-neg-in 63.3%

      \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(--1\right)}}{2} \]

   9. metadata-eval 63.3%

      \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \color{blue}{1}}{2} \]

   10. *-rgt-identity 63.3%

       \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}}{2} \]

   11. +-commutative 63.3%

       \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\color{blue}{\left(1 + \varepsilon\right)}}}{2} \]

   12. exp-prod 68.6%

       \[\leadsto \frac{1 + \color{blue}{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}}{2} \]

   13. distribute-lft-in 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot 1 + \left(-x\right) \cdot \varepsilon}}}{2} \]

   14. *-rgt-identity 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right)} + \left(-x\right) \cdot \varepsilon}}{2} \]

   15. cancel-sign-sub-inv 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) - x \cdot \varepsilon}}}{2} \]

   16. neg-mul-1 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{-1 \cdot x} - x \cdot \varepsilon}}{2} \]

   17. *-commutative 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{x \cdot -1} - x \cdot \varepsilon}}{2} \]

   18. distribute-lft-out-- 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

10. Simplified 68.6%

    \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

11. Taylor expanded in x around 0 76.6%

    \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}}{2} \]
12. Step-by-step derivation

    1. distribute-lft-in 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    2. metadata-eval 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    3. neg-mul-1 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    4. sub-neg 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 - \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    5. +-commutative 76.6%

       \[\leadsto \frac{2 + x \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

13. Simplified 76.6%

    \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

14. Taylor expanded in eps around 0 77.3%

    \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{\left(x + \varepsilon \cdot \left(2 \cdot x + \varepsilon \cdot x\right)\right)} + \left(-1 - \varepsilon\right)\right)}{2} \]
15. Step-by-step derivation

    1. distribute-rgt-out 77.3%

       \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \left(x + \varepsilon \cdot \color{blue}{\left(x \cdot \left(2 + \varepsilon\right)\right)}\right) + \left(-1 - \varepsilon\right)\right)}{2} \]

16. Simplified 77.3%

    \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \color{blue}{\left(x + \varepsilon \cdot \left(x \cdot \left(2 + \varepsilon\right)\right)\right)} + \left(-1 - \varepsilon\right)\right)}{2} \]

17. Final simplification 77.3%

    \[\leadsto \frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x + \varepsilon \cdot \left(x \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2} \]
18. Add Preprocessing

Alternative 18: 74.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (+ 2.0 (* x (+ (- -1.0 eps) (* 0.5 (* x (+ 1.0 (* eps (+ eps 2.0))))))))
  2.0))

C

double code(double x, double eps) {
	return (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 + (x * (((-1.0d0) - eps) + (0.5d0 * (x * (1.0d0 + (eps * (eps + 2.0d0)))))))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0;
}

Python

def code(x, eps):
	return (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 - eps) + Float64(0.5 * Float64(x * Float64(1.0 + Float64(eps * Float64(eps + 2.0)))))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 + (x * ((-1.0 - eps) + (0.5 * (x * (1.0 + (eps * (eps + 2.0)))))))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 + N[(x * N[(N[(-1.0 - eps), $MachinePrecision] + N[(0.5 * N[(x * N[(1.0 + N[(eps * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 72.0%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 41.2%

   \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
6. Step-by-step derivation

   1. metadata-eval 41.2%

      \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   2. distribute-neg-frac 41.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   3. metadata-eval 41.2%

      \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   4. associate-*l/ 41.2%

      \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. *-commutative 41.2%

      \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. distribute-lft-neg-in 41.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. cancel-sign-sub-inv 41.2%

      \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. *-commutative 41.2%

      \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   9. associate-*l/ 41.2%

      \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   10. metadata-eval 41.2%

       \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

7. Simplified 41.2%

   \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

8. Taylor expanded in eps around inf 68.6%

   \[\leadsto \frac{\color{blue}{1 - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
9. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \frac{1 - \color{blue}{e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot -1}}{2} \]

   2. associate-*r* 68.6%

      \[\leadsto \frac{1 - e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}} \cdot -1}{2} \]

   3. neg-mul-1 68.6%

      \[\leadsto \frac{1 - e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)} \cdot -1}{2} \]

   4. exp-prod 63.3%

      \[\leadsto \frac{1 - \color{blue}{{\left(e^{-x}\right)}^{\left(1 + \varepsilon\right)}} \cdot -1}{2} \]

   5. +-commutative 63.3%

      \[\leadsto \frac{1 - {\left(e^{-x}\right)}^{\color{blue}{\left(\varepsilon + 1\right)}} \cdot -1}{2} \]

   6. cancel-sign-sub-inv 63.3%

      \[\leadsto \frac{\color{blue}{1 + \left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}\right) \cdot -1}}{2} \]

   7. distribute-lft-neg-in 63.3%

      \[\leadsto \frac{1 + \color{blue}{\left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot -1\right)}}{2} \]

   8. distribute-rgt-neg-in 63.3%

      \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(--1\right)}}{2} \]

   9. metadata-eval 63.3%

      \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \color{blue}{1}}{2} \]

   10. *-rgt-identity 63.3%

       \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}}{2} \]

   11. +-commutative 63.3%

       \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\color{blue}{\left(1 + \varepsilon\right)}}}{2} \]

   12. exp-prod 68.6%

       \[\leadsto \frac{1 + \color{blue}{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}}{2} \]

   13. distribute-lft-in 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot 1 + \left(-x\right) \cdot \varepsilon}}}{2} \]

   14. *-rgt-identity 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right)} + \left(-x\right) \cdot \varepsilon}}{2} \]

   15. cancel-sign-sub-inv 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) - x \cdot \varepsilon}}}{2} \]

   16. neg-mul-1 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{-1 \cdot x} - x \cdot \varepsilon}}{2} \]

   17. *-commutative 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{x \cdot -1} - x \cdot \varepsilon}}{2} \]

   18. distribute-lft-out-- 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

10. Simplified 68.6%

    \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

11. Taylor expanded in x around 0 76.6%

    \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}}{2} \]
12. Step-by-step derivation

    1. distribute-lft-in 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    2. metadata-eval 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    3. neg-mul-1 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    4. sub-neg 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 - \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    5. +-commutative 76.6%

       \[\leadsto \frac{2 + x \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

13. Simplified 76.6%

    \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

14. Taylor expanded in eps around 0 76.6%

    \[\leadsto \frac{2 + x \cdot \left(0.5 \cdot \left(x \cdot \color{blue}{\left(1 + \varepsilon \cdot \left(2 + \varepsilon\right)\right)}\right) + \left(-1 - \varepsilon\right)\right)}{2} \]

15. Final simplification 76.6%

    \[\leadsto \frac{2 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(x \cdot \left(1 + \varepsilon \cdot \left(\varepsilon + 2\right)\right)\right)\right)}{2} \]
16. Add Preprocessing

Alternative 19: 52.1% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \frac{2 + x \cdot \left(-1 + \left(x \cdot 0.5 + \varepsilon \cdot \left(x + -1\right)\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (+ 2.0 (* x (+ -1.0 (+ (* x 0.5) (* eps (+ x -1.0)))))) 2.0))

C

double code(double x, double eps) {
	return (2.0 + (x * (-1.0 + ((x * 0.5) + (eps * (x + -1.0)))))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 + (x * ((-1.0d0) + ((x * 0.5d0) + (eps * (x + (-1.0d0))))))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 + (x * (-1.0 + ((x * 0.5) + (eps * (x + -1.0)))))) / 2.0;
}

Python

def code(x, eps):
	return (2.0 + (x * (-1.0 + ((x * 0.5) + (eps * (x + -1.0)))))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(Float64(x * 0.5) + Float64(eps * Float64(x + -1.0)))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 + (x * (-1.0 + ((x * 0.5) + (eps * (x + -1.0)))))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 + N[(x * N[(-1.0 + N[(N[(x * 0.5), $MachinePrecision] + N[(eps * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 72.0%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 41.2%

   \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
6. Step-by-step derivation

   1. metadata-eval 41.2%

      \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   2. distribute-neg-frac 41.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   3. metadata-eval 41.2%

      \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   4. associate-*l/ 41.2%

      \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   5. *-commutative 41.2%

      \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. distribute-lft-neg-in 41.2%

      \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. cancel-sign-sub-inv 41.2%

      \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. *-commutative 41.2%

      \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   9. associate-*l/ 41.2%

      \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   10. metadata-eval 41.2%

       \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

7. Simplified 41.2%

   \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

8. Taylor expanded in eps around inf 68.6%

   \[\leadsto \frac{\color{blue}{1 - -1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
9. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \frac{1 - \color{blue}{e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \cdot -1}}{2} \]

   2. associate-*r* 68.6%

      \[\leadsto \frac{1 - e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 + \varepsilon\right)}} \cdot -1}{2} \]

   3. neg-mul-1 68.6%

      \[\leadsto \frac{1 - e^{\color{blue}{\left(-x\right)} \cdot \left(1 + \varepsilon\right)} \cdot -1}{2} \]

   4. exp-prod 63.3%

      \[\leadsto \frac{1 - \color{blue}{{\left(e^{-x}\right)}^{\left(1 + \varepsilon\right)}} \cdot -1}{2} \]

   5. +-commutative 63.3%

      \[\leadsto \frac{1 - {\left(e^{-x}\right)}^{\color{blue}{\left(\varepsilon + 1\right)}} \cdot -1}{2} \]

   6. cancel-sign-sub-inv 63.3%

      \[\leadsto \frac{\color{blue}{1 + \left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}\right) \cdot -1}}{2} \]

   7. distribute-lft-neg-in 63.3%

      \[\leadsto \frac{1 + \color{blue}{\left(-{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot -1\right)}}{2} \]

   8. distribute-rgt-neg-in 63.3%

      \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \left(--1\right)}}{2} \]

   9. metadata-eval 63.3%

      \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)} \cdot \color{blue}{1}}{2} \]

   10. *-rgt-identity 63.3%

       \[\leadsto \frac{1 + \color{blue}{{\left(e^{-x}\right)}^{\left(\varepsilon + 1\right)}}}{2} \]

   11. +-commutative 63.3%

       \[\leadsto \frac{1 + {\left(e^{-x}\right)}^{\color{blue}{\left(1 + \varepsilon\right)}}}{2} \]

   12. exp-prod 68.6%

       \[\leadsto \frac{1 + \color{blue}{e^{\left(-x\right) \cdot \left(1 + \varepsilon\right)}}}{2} \]

   13. distribute-lft-in 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) \cdot 1 + \left(-x\right) \cdot \varepsilon}}}{2} \]

   14. *-rgt-identity 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right)} + \left(-x\right) \cdot \varepsilon}}{2} \]

   15. cancel-sign-sub-inv 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{\left(-x\right) - x \cdot \varepsilon}}}{2} \]

   16. neg-mul-1 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{-1 \cdot x} - x \cdot \varepsilon}}{2} \]

   17. *-commutative 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{x \cdot -1} - x \cdot \varepsilon}}{2} \]

   18. distribute-lft-out-- 68.6%

       \[\leadsto \frac{1 + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

10. Simplified 68.6%

    \[\leadsto \frac{\color{blue}{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

11. Taylor expanded in x around 0 76.6%

    \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}}{2} \]
12. Step-by-step derivation

    1. distribute-lft-in 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    2. metadata-eval 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\left(\color{blue}{-1} + -1 \cdot \varepsilon\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    3. neg-mul-1 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\left(-1 + \color{blue}{\left(-\varepsilon\right)}\right) + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    4. sub-neg 76.6%

       \[\leadsto \frac{2 + x \cdot \left(\color{blue}{\left(-1 - \varepsilon\right)} + 0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}{2} \]

    5. +-commutative 76.6%

       \[\leadsto \frac{2 + x \cdot \color{blue}{\left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

13. Simplified 76.6%

    \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right) + \left(-1 - \varepsilon\right)\right)}}{2} \]

14. Taylor expanded in eps around 0 61.8%

    \[\leadsto \frac{2 + x \cdot \color{blue}{\left(\left(0.5 \cdot x + \varepsilon \cdot \left(x - 1\right)\right) - 1\right)}}{2} \]

15. Final simplification 61.8%

    \[\leadsto \frac{2 + x \cdot \left(-1 + \left(x \cdot 0.5 + \varepsilon \cdot \left(x + -1\right)\right)\right)}{2} \]
16. Add Preprocessing

Alternative 20: 59.6% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 280:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 280.0) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 280.0) {
		tmp = (2.0 - (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 <= 280.0d0) then
        tmp = (2.0d0 - (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 <= 280.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 280.0:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 280.0)
		tmp = Float64(Float64(2.0 - 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 <= 280.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 280.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 280

   1. Initial program 62.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 48.3%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 48.3%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 48.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 48.3%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 48.3%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 48.3%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 48.3%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 48.3%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 48.3%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 48.3%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 48.3%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 48.3%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in x around 0 54.3%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2} \]

   9. Taylor expanded in eps around inf 73.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(-1 \cdot x + 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]
   10. Step-by-step derivation

       1. neg-mul-1 73.4%

          \[\leadsto \frac{\varepsilon \cdot \left(\color{blue}{\left(-x\right)} + 2 \cdot \frac{1}{\varepsilon}\right)}{2} \]

       2. +-commutative 73.4%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(2 \cdot \frac{1}{\varepsilon} + \left(-x\right)\right)}}{2} \]

       3. distribute-lft-in 73.4%

          \[\leadsto \frac{\color{blue}{\varepsilon \cdot \left(2 \cdot \frac{1}{\varepsilon}\right) + \varepsilon \cdot \left(-x\right)}}{2} \]

       4. *-commutative 73.4%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\frac{1}{\varepsilon} \cdot 2\right)} + \varepsilon \cdot \left(-x\right)}{2} \]

       5. associate-*r* 73.4%

          \[\leadsto \frac{\color{blue}{\left(\varepsilon \cdot \frac{1}{\varepsilon}\right) \cdot 2} + \varepsilon \cdot \left(-x\right)}{2} \]

       6. rgt-mult-inverse 73.5%

          \[\leadsto \frac{\color{blue}{1} \cdot 2 + \varepsilon \cdot \left(-x\right)}{2} \]

       7. metadata-eval 73.5%

          \[\leadsto \frac{\color{blue}{2} + \varepsilon \cdot \left(-x\right)}{2} \]

       8. distribute-rgt-neg-in 73.5%

          \[\leadsto \frac{2 + \color{blue}{\left(-\varepsilon \cdot x\right)}}{2} \]

       9. unsub-neg 73.5%

          \[\leadsto \frac{\color{blue}{2 - \varepsilon \cdot x}}{2} \]

       10. *-commutative 73.5%

           \[\leadsto \frac{2 - \color{blue}{x \cdot \varepsilon}}{2} \]

   11. Simplified 73.5%

       \[\leadsto \frac{\color{blue}{2 - x \cdot \varepsilon}}{2} \]

   if 280 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 50.0%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

      2. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 50.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 50.0%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 50.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 50.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 50.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 280:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 60.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\varepsilon \cdot x}{-2}\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0) (/ (* eps x) (- 2.0)) (if (<= x 490.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * x) / -2.0;
	} else if (x <= 490.0) {
		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 <= (-1.0d0)) then
        tmp = (eps * x) / -2.0d0
    else if (x <= 490.0d0) 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 <= -1.0) {
		tmp = (eps * x) / -2.0;
	} else if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (eps * x) / -2.0
	elif x <= 490.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(eps * x) / Float64(-2.0));
	elseif (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (eps * x) / -2.0;
	elseif (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(eps * x), $MachinePrecision] / (-2.0)), $MachinePrecision], If[LessEqual[x, 490.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   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. Add Preprocessing

   5. Taylor expanded in x around 0 70.9%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
   6. Step-by-step derivation

      1. metadata-eval 70.9%

         \[\leadsto \frac{\left(1 + \frac{\color{blue}{--1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      2. distribute-neg-frac 70.9%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(-\frac{-1}{\varepsilon}\right)}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      3. metadata-eval 70.9%

         \[\leadsto \frac{\left(1 + \left(-\frac{\color{blue}{1 \cdot -1}}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      4. associate-*l/ 70.9%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{\frac{1}{\varepsilon} \cdot -1}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      5. *-commutative 70.9%

         \[\leadsto \frac{\left(1 + \left(-\color{blue}{-1 \cdot \frac{1}{\varepsilon}}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      6. distribute-lft-neg-in 70.9%

         \[\leadsto \frac{\left(1 + \color{blue}{\left(--1\right) \cdot \frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      7. cancel-sign-sub-inv 70.9%

         \[\leadsto \frac{\color{blue}{\left(1 - -1 \cdot \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      8. *-commutative 70.9%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1}{\varepsilon} \cdot -1}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      9. associate-*l/ 70.9%

         \[\leadsto \frac{\left(1 - \color{blue}{\frac{1 \cdot -1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

      10. metadata-eval 70.9%

          \[\leadsto \frac{\left(1 - \frac{\color{blue}{-1}}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Simplified 70.9%

      \[\leadsto \frac{\color{blue}{\left(1 - \frac{-1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   8. Taylor expanded in x around 0 51.1%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}}{2} \]

   9. Taylor expanded in eps around -inf 51.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\varepsilon \cdot \left(x - 2 \cdot \frac{1}{\varepsilon}\right)\right)}}{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 51.1%

          \[\leadsto \frac{\color{blue}{-\varepsilon \cdot \left(x - 2 \cdot \frac{1}{\varepsilon}\right)}}{2} \]

       2. *-commutative 51.1%

          \[\leadsto \frac{-\color{blue}{\left(x - 2 \cdot \frac{1}{\varepsilon}\right) \cdot \varepsilon}}{2} \]

       3. distribute-rgt-neg-in 51.1%

          \[\leadsto \frac{\color{blue}{\left(x - 2 \cdot \frac{1}{\varepsilon}\right) \cdot \left(-\varepsilon\right)}}{2} \]

       4. sub-neg 51.1%

          \[\leadsto \frac{\color{blue}{\left(x + \left(-2 \cdot \frac{1}{\varepsilon}\right)\right)} \cdot \left(-\varepsilon\right)}{2} \]

       5. associate-*r/ 51.1%

          \[\leadsto \frac{\left(x + \left(-\color{blue}{\frac{2 \cdot 1}{\varepsilon}}\right)\right) \cdot \left(-\varepsilon\right)}{2} \]

       6. metadata-eval 51.1%

          \[\leadsto \frac{\left(x + \left(-\frac{\color{blue}{2}}{\varepsilon}\right)\right) \cdot \left(-\varepsilon\right)}{2} \]

       7. distribute-neg-frac 51.1%

          \[\leadsto \frac{\left(x + \color{blue}{\frac{-2}{\varepsilon}}\right) \cdot \left(-\varepsilon\right)}{2} \]

       8. metadata-eval 51.1%

          \[\leadsto \frac{\left(x + \frac{\color{blue}{-2}}{\varepsilon}\right) \cdot \left(-\varepsilon\right)}{2} \]

   11. Simplified 51.1%

       \[\leadsto \frac{\color{blue}{\left(x + \frac{-2}{\varepsilon}\right) \cdot \left(-\varepsilon\right)}}{2} \]

   12. Taylor expanded in x around inf 51.1%

       \[\leadsto \frac{\color{blue}{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
   13. Step-by-step derivation

       1. associate-*r* 51.1%

          \[\leadsto \frac{\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot x}}{2} \]

       2. mul-1-neg 51.1%

          \[\leadsto \frac{\color{blue}{\left(-\varepsilon\right)} \cdot x}{2} \]

   14. Simplified 51.1%

       \[\leadsto \frac{\color{blue}{\left(-\varepsilon\right) \cdot x}}{2} \]

   if -1 < x < 490

   1. Initial program 56.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 78.5%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 490 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 50.0%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

      2. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 50.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 50.0%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 50.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 50.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 50.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\varepsilon \cdot x}{-2}\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 56.9% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 620.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 620.0) {
		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 <= 620.0d0) 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 <= 620.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 620.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 620.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 620.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 620.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 620

   1. Initial program 62.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.7%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 620 < 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%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 50.0%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
   6. Step-by-step derivation

      1. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

      2. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

      3. rec-exp 50.0%

         \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

      4. sub-neg 50.0%

         \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

      5. div-sub 50.0%

         \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

      6. mul-1-neg 50.0%

         \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      7. rec-exp 50.0%

         \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

      8. +-inverses 50.0%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 50.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 16.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 72.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

3. Simplified 54.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
4. Add Preprocessing

5. Taylor expanded in eps around 0 14.1%

   \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x} + -1 \cdot e^{-1 \cdot x}}{\varepsilon}}}{2} \]
6. Step-by-step derivation

   1. mul-1-neg 14.1%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} + \color{blue}{\left(-e^{-1 \cdot x}\right)}}{\varepsilon}}{2} \]

   2. mul-1-neg 14.1%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-e^{\color{blue}{-x}}\right)}{\varepsilon}}{2} \]

   3. rec-exp 14.1%

      \[\leadsto \frac{\frac{e^{-1 \cdot x} + \left(-\color{blue}{\frac{1}{e^{x}}}\right)}{\varepsilon}}{2} \]

   4. sub-neg 14.1%

      \[\leadsto \frac{\frac{\color{blue}{e^{-1 \cdot x} - \frac{1}{e^{x}}}}{\varepsilon}}{2} \]

   5. div-sub 14.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]

   6. mul-1-neg 14.1%

      \[\leadsto \frac{\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

   7. rec-exp 14.1%

      \[\leadsto \frac{\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}{2} \]

   8. +-inverses 14.3%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Simplified 14.3%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

8. Final simplification 14.3%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))