NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.8%

Time: 17.1s

Alternatives: 22

Speedup: 9.4×

• 37.6% 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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left({\left(e^{x \cdot \frac{\frac{eps\_m}{2}}{2}}\right)}^{2}, e^{x \cdot \frac{eps\_m}{2}}, e^{x \cdot \left(-1 - eps\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00084)
   (* (+ x 1.0) (exp (- 0.0 x)))
   (*
    0.5
    (fma
     (pow (exp (* x (/ (/ eps_m 2.0) 2.0))) 2.0)
     (exp (* x (/ eps_m 2.0)))
     (exp (* x (- -1.0 eps_m)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = 0.5 * fma(pow(exp((x * ((eps_m / 2.0) / 2.0))), 2.0), exp((x * (eps_m / 2.0))), exp((x * (-1.0 - eps_m))));
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(0.5 * fma((exp(Float64(x * Float64(Float64(eps_m / 2.0) / 2.0))) ^ 2.0), exp(Float64(x * Float64(eps_m / 2.0))), exp(Float64(x * Float64(-1.0 - eps_m)))));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00084], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Power[N[Exp[N[(x * N[(N[(eps$95$m / 2.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 68.6%

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

   if 8.4000000000000003e-4 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

       1. exp-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(e^{x}\right)}^{\varepsilon} + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)} + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}\right)\right) \]

       3. fma-define N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\mathsf{fma}\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}, \color{blue}{{\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}}, e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       4. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right)}, \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       5. pow-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left(e^{x \cdot \frac{\varepsilon}{2}}\right), \left({\color{blue}{\left(e^{x}\right)}}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       6. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \frac{\varepsilon}{2}\right)\right), \left({\color{blue}{\left(e^{x}\right)}}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\varepsilon}{2}\right)\right)\right), \left({\left(e^{\color{blue}{x}}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       9. pow-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \left(e^{x \cdot \frac{\varepsilon}{2}}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       10. exp-lowering-exp.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \frac{\varepsilon}{2}\right)\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\varepsilon}{2}\right)\right)\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       13. exp-lowering-exp.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

       15. --lowering--.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(e^{x \cdot \frac{\varepsilon}{2}}, e^{x \cdot \frac{\varepsilon}{2}}, e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \]
   13. Step-by-step derivation

       1. exp-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left({\left(e^{x}\right)}^{\left(\frac{\frac{\varepsilon}{2}}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\frac{\varepsilon}{2}}{2}\right)}\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left({\left({\left(e^{x}\right)}^{\left(\frac{\frac{\varepsilon}{2}}{2}\right)}\right)}^{2}\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(\left({\left(e^{x}\right)}^{\left(\frac{\frac{\varepsilon}{2}}{2}\right)}\right), 2\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       5. pow-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(\left(e^{x \cdot \frac{\frac{\varepsilon}{2}}{2}}\right), 2\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       6. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \frac{\frac{\varepsilon}{2}}{2}\right)\right), 2\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{\varepsilon}{2}}{2}\right)\right)\right), 2\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{\varepsilon}{2}\right), 2\right)\right)\right), 2\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{pow.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\varepsilon, 2\right), 2\right)\right)\right), 2\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 100.0%

       \[\leadsto 0.5 \cdot \mathsf{fma}\left(\color{blue}{{\left(e^{x \cdot \frac{\frac{\varepsilon}{2}}{2}}\right)}^{2}}, e^{x \cdot \frac{\varepsilon}{2}}, e^{x \cdot \left(-1 - \varepsilon\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \frac{eps\_m}{2}}\\ \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(t\_0, t\_0, e^{x \cdot \left(-1 - eps\_m\right)}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (/ eps_m 2.0)))))
   (if (<= eps_m 0.00084)
     (* (+ x 1.0) (exp (- 0.0 x)))
     (* 0.5 (fma t_0 t_0 (exp (* x (- -1.0 eps_m))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m / 2.0)));
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = 0.5 * fma(t_0, t_0, exp((x * (-1.0 - eps_m))));
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m / 2.0)))
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(0.5 * fma(t_0, t_0, exp(Float64(x * Float64(-1.0 - eps_m)))));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.00084], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 * t$95$0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 68.6%

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

   if 8.4000000000000003e-4 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

       1. exp-prod N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(e^{x}\right)}^{\varepsilon} + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}\right)\right) \]

       2. sqr-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)} + e^{\color{blue}{x \cdot \left(-1 - \varepsilon\right)}}\right)\right) \]

       3. fma-define N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \left(\mathsf{fma}\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}, \color{blue}{{\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}}, e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       4. fma-lowering-fma.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right)}, \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       5. pow-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\left(e^{x \cdot \frac{\varepsilon}{2}}\right), \left({\color{blue}{\left(e^{x}\right)}}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       6. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \frac{\varepsilon}{2}\right)\right), \left({\color{blue}{\left(e^{x}\right)}}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\varepsilon}{2}\right)\right)\right), \left({\left(e^{\color{blue}{x}}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \left({\left(e^{x}\right)}^{\left(\frac{\varepsilon}{2}\right)}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       9. pow-exp N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \left(e^{x \cdot \frac{\varepsilon}{2}}\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       10. exp-lowering-exp.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \frac{\varepsilon}{2}\right)\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\varepsilon}{2}\right)\right)\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \left(e^{x \cdot \left(-1 - \varepsilon\right)}\right)\right)\right) \]

       13. exp-lowering-exp.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

       15. --lowering--.f64 100.0%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{fma.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\varepsilon, 2\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(e^{x \cdot \frac{\varepsilon}{2}}, e^{x \cdot \frac{\varepsilon}{2}}, e^{x \cdot \left(-1 - \varepsilon\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - eps\_m\right)} + e^{eps\_m \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00084)
   (* (+ x 1.0) (exp (- 0.0 x)))
   (* 0.5 (+ (exp (* x (- -1.0 eps_m))) (exp (* eps_m x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((eps_m * x)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.00084d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else
        tmp = 0.5d0 * (exp((x * ((-1.0d0) - eps_m))) + exp((eps_m * x)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else {
		tmp = 0.5 * (Math.exp((x * (-1.0 - eps_m))) + Math.exp((eps_m * x)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00084:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	else:
		tmp = 0.5 * (math.exp((x * (-1.0 - eps_m))) + math.exp((eps_m * x)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + exp(Float64(eps_m * x))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00084)
		tmp = (x + 1.0) * exp((0.0 - x));
	else
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((eps_m * x)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00084], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 68.6%

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

   if 8.4000000000000003e-4 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\left(eps\_m + -1\right) \cdot \left(1 + \left(x \cdot 0.5\right) \cdot \left(eps\_m + -1\right)\right)\right)\right) \cdot \left(0.5 - \frac{-0.5}{eps\_m}\right) - e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(-0.5 + \frac{0.5}{eps\_m}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00084)
   (* (+ x 1.0) (exp (- 0.0 x)))
   (-
    (*
     (+ 1.0 (* x (* (+ eps_m -1.0) (+ 1.0 (* (* x 0.5) (+ eps_m -1.0))))))
     (- 0.5 (/ -0.5 eps_m)))
    (* (exp (* x (- -1.0 eps_m))) (+ -0.5 (/ 0.5 eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = ((1.0 + (x * ((eps_m + -1.0) * (1.0 + ((x * 0.5) * (eps_m + -1.0)))))) * (0.5 - (-0.5 / eps_m))) - (exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.00084d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else
        tmp = ((1.0d0 + (x * ((eps_m + (-1.0d0)) * (1.0d0 + ((x * 0.5d0) * (eps_m + (-1.0d0))))))) * (0.5d0 - ((-0.5d0) / eps_m))) - (exp((x * ((-1.0d0) - eps_m))) * ((-0.5d0) + (0.5d0 / eps_m)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else {
		tmp = ((1.0 + (x * ((eps_m + -1.0) * (1.0 + ((x * 0.5) * (eps_m + -1.0)))))) * (0.5 - (-0.5 / eps_m))) - (Math.exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00084:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	else:
		tmp = ((1.0 + (x * ((eps_m + -1.0) * (1.0 + ((x * 0.5) * (eps_m + -1.0)))))) * (0.5 - (-0.5 / eps_m))) - (math.exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * Float64(Float64(eps_m + -1.0) * Float64(1.0 + Float64(Float64(x * 0.5) * Float64(eps_m + -1.0)))))) * Float64(0.5 - Float64(-0.5 / eps_m))) - Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(-0.5 + Float64(0.5 / eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00084)
		tmp = (x + 1.0) * exp((0.0 - x));
	else
		tmp = ((1.0 + (x * ((eps_m + -1.0) * (1.0 + ((x * 0.5) * (eps_m + -1.0)))))) * (0.5 - (-0.5 / eps_m))) - (exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00084], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x * N[(N[(eps$95$m + -1.0), $MachinePrecision] * N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(-0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 68.6%

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

   if 8.4000000000000003e-4 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + \frac{1}{2} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\varepsilon + \frac{1}{2} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + \frac{1}{2} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(-1, \varepsilon\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right) + \varepsilon\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right) + \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot x\right) \cdot {\left(\varepsilon - 1\right)}^{2} + \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\left(\varepsilon - 1\right) \cdot \left(\varepsilon - 1\right)\right) + \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon - 1\right)\right) \cdot \left(\varepsilon - 1\right) + \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. distribute-lft1-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon - 1\right) + 1\right) \cdot \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon - 1\right) + 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot x\right) \cdot \left(\varepsilon - 1\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot x\right), \left(\varepsilon - 1\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \frac{1}{2}\right), \left(\varepsilon - 1\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\varepsilon - 1\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \left(\varepsilon + -1\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{+.f64}\left(\varepsilon, -1\right)\right), 1\right), \left(\varepsilon - 1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{+.f64}\left(\varepsilon, -1\right)\right), 1\right), \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{+.f64}\left(\varepsilon, -1\right)\right), 1\right), \left(\varepsilon + -1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      19. +-lowering-+.f64 83.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{2}\right), \mathsf{+.f64}\left(\varepsilon, -1\right)\right), 1\right), \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 83.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(1 + \left(x \cdot 0.5\right) \cdot \left(\varepsilon + -1\right)\right)\right)\right) \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(-0.5 + \frac{0.5}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\\ \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{elif}\;eps\_m \leq 5.6 \cdot 10^{+47}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{eps\_m}\right) - e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(-0.5 + \frac{0.5}{eps\_m}\right)\\ \mathbf{elif}\;eps\_m \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(eps\_m + 1\right) + \frac{t\_0 \cdot \left(1 + eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)}{1 + eps\_m \cdot \left(eps\_m + -1\right)}\right)\\ \mathbf{elif}\;eps\_m \leq 2.7 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* (+ eps_m 1.0) 0.25))))
   (if (<= eps_m 0.00084)
     (* (+ x 1.0) (exp (- 0.0 x)))
     (if (<= eps_m 5.6e+47)
       (-
        (+ 0.5 (/ 0.5 eps_m))
        (* (exp (* x (- -1.0 eps_m))) (+ -0.5 (/ 0.5 eps_m))))
       (if (<= eps_m 4.2e+125)
         (+
          1.0
          (*
           x
           (+
            (* -0.5 (+ eps_m 1.0))
            (/
             (* t_0 (+ 1.0 (* eps_m (* eps_m eps_m))))
             (+ 1.0 (* eps_m (+ eps_m -1.0)))))))
         (if (<= eps_m 2.7e+184)
           (+
            1.0
            (*
             x
             (*
              0.5
              (+
               (+ eps_m (- -1.0 eps_m))
               (*
                x
                (*
                 0.5
                 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
           (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 t_0))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else if (eps_m <= 5.6e+47) {
		tmp = (0.5 + (0.5 / eps_m)) - (exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	} else if (eps_m <= 4.2e+125) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 2.7e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((eps_m + 1.0d0) * 0.25d0)
    if (eps_m <= 0.00084d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else if (eps_m <= 5.6d+47) then
        tmp = (0.5d0 + (0.5d0 / eps_m)) - (exp((x * ((-1.0d0) - eps_m))) * ((-0.5d0) + (0.5d0 / eps_m)))
    else if (eps_m <= 4.2d+125) then
        tmp = 1.0d0 + (x * (((-0.5d0) * (eps_m + 1.0d0)) + ((t_0 * (1.0d0 + (eps_m * (eps_m * eps_m)))) / (1.0d0 + (eps_m * (eps_m + (-1.0d0)))))))
    else if (eps_m <= 2.7d+184) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))
    else
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + t_0)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else if (eps_m <= 5.6e+47) {
		tmp = (0.5 + (0.5 / eps_m)) - (Math.exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	} else if (eps_m <= 4.2e+125) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 2.7e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * ((eps_m + 1.0) * 0.25)
	tmp = 0
	if eps_m <= 0.00084:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	elif eps_m <= 5.6e+47:
		tmp = (0.5 + (0.5 / eps_m)) - (math.exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)))
	elif eps_m <= 4.2e+125:
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))))
	elif eps_m <= 2.7e+184:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))))
	else:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	elseif (eps_m <= 5.6e+47)
		tmp = Float64(Float64(0.5 + Float64(0.5 / eps_m)) - Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(-0.5 + Float64(0.5 / eps_m))));
	elseif (eps_m <= 4.2e+125)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-0.5 * Float64(eps_m + 1.0)) + Float64(Float64(t_0 * Float64(1.0 + Float64(eps_m * Float64(eps_m * eps_m)))) / Float64(1.0 + Float64(eps_m * Float64(eps_m + -1.0)))))));
	elseif (eps_m <= 2.7e+184)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + t_0))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * ((eps_m + 1.0) * 0.25);
	tmp = 0.0;
	if (eps_m <= 0.00084)
		tmp = (x + 1.0) * exp((0.0 - x));
	elseif (eps_m <= 5.6e+47)
		tmp = (0.5 + (0.5 / eps_m)) - (exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	elseif (eps_m <= 4.2e+125)
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	elseif (eps_m <= 2.7e+184)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	else
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.00084], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 5.6e+47], N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 4.2e+125], N[(1.0 + N[(x * N[(N[(-0.5 * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(1.0 + N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 2.7e+184], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 68.6%

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

   if 8.4000000000000003e-4 < eps < 5.59999999999999976e47

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 83.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 83.8%

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

   if 5.59999999999999976e47 < eps < 4.2000000000000001e125

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 43.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 43.5%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 43.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 52.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 52.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \frac{{1}^{3} + {\varepsilon}^{3}}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1 \cdot 1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \frac{1}{4}\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon + 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left(1 + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{3}\right)\right)\right), \left(1 \cdot \color{blue}{1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 + \left(\color{blue}{\varepsilon \cdot \varepsilon} - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

       19. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 91.9%

       \[\leadsto 1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \color{blue}{\frac{\left(x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon - 1\right)}}\right) \]

   if 4.2000000000000001e125 < eps < 2.6999999999999999e184

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 81.2%

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

   if 2.6999999999999999e184 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 45.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 45.9%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 45.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 87.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 87.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 91.2%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{elif}\;\varepsilon \leq 5.6 \cdot 10^{+47}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(-0.5 + \frac{0.5}{\varepsilon}\right)\\ \mathbf{elif}\;\varepsilon \leq 4.2 \cdot 10^{+125}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + \frac{\left(x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{elif}\;\varepsilon \leq 2.7 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\\ \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{elif}\;eps\_m \leq 6.2 \cdot 10^{+47}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - eps\_m\right)}\\ \mathbf{elif}\;eps\_m \leq 2 \cdot 10^{+126}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(eps\_m + 1\right) + \frac{t\_0 \cdot \left(1 + eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)}{1 + eps\_m \cdot \left(eps\_m + -1\right)}\right)\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* (+ eps_m 1.0) 0.25))))
   (if (<= eps_m 0.00084)
     (* (+ x 1.0) (exp (- 0.0 x)))
     (if (<= eps_m 6.2e+47)
       (+ 0.5 (* 0.5 (exp (* x (- -1.0 eps_m)))))
       (if (<= eps_m 2e+126)
         (+
          1.0
          (*
           x
           (+
            (* -0.5 (+ eps_m 1.0))
            (/
             (* t_0 (+ 1.0 (* eps_m (* eps_m eps_m))))
             (+ 1.0 (* eps_m (+ eps_m -1.0)))))))
         (if (<= eps_m 1.55e+184)
           (+
            1.0
            (*
             x
             (*
              0.5
              (+
               (+ eps_m (- -1.0 eps_m))
               (*
                x
                (*
                 0.5
                 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
           (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 t_0))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else if (eps_m <= 6.2e+47) {
		tmp = 0.5 + (0.5 * exp((x * (-1.0 - eps_m))));
	} else if (eps_m <= 2e+126) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 1.55e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((eps_m + 1.0d0) * 0.25d0)
    if (eps_m <= 0.00084d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else if (eps_m <= 6.2d+47) then
        tmp = 0.5d0 + (0.5d0 * exp((x * ((-1.0d0) - eps_m))))
    else if (eps_m <= 2d+126) then
        tmp = 1.0d0 + (x * (((-0.5d0) * (eps_m + 1.0d0)) + ((t_0 * (1.0d0 + (eps_m * (eps_m * eps_m)))) / (1.0d0 + (eps_m * (eps_m + (-1.0d0)))))))
    else if (eps_m <= 1.55d+184) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))
    else
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + t_0)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else if (eps_m <= 6.2e+47) {
		tmp = 0.5 + (0.5 * Math.exp((x * (-1.0 - eps_m))));
	} else if (eps_m <= 2e+126) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 1.55e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * ((eps_m + 1.0) * 0.25)
	tmp = 0
	if eps_m <= 0.00084:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	elif eps_m <= 6.2e+47:
		tmp = 0.5 + (0.5 * math.exp((x * (-1.0 - eps_m))))
	elif eps_m <= 2e+126:
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))))
	elif eps_m <= 1.55e+184:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))))
	else:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	elseif (eps_m <= 6.2e+47)
		tmp = Float64(0.5 + Float64(0.5 * exp(Float64(x * Float64(-1.0 - eps_m)))));
	elseif (eps_m <= 2e+126)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-0.5 * Float64(eps_m + 1.0)) + Float64(Float64(t_0 * Float64(1.0 + Float64(eps_m * Float64(eps_m * eps_m)))) / Float64(1.0 + Float64(eps_m * Float64(eps_m + -1.0)))))));
	elseif (eps_m <= 1.55e+184)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + t_0))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * ((eps_m + 1.0) * 0.25);
	tmp = 0.0;
	if (eps_m <= 0.00084)
		tmp = (x + 1.0) * exp((0.0 - x));
	elseif (eps_m <= 6.2e+47)
		tmp = 0.5 + (0.5 * exp((x * (-1.0 - eps_m))));
	elseif (eps_m <= 2e+126)
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	elseif (eps_m <= 1.55e+184)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	else
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.00084], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 6.2e+47], N[(0.5 + N[(0.5 * N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 2e+126], N[(1.0 + N[(x * N[(N[(-0.5 * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(1.0 + N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.55e+184], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 68.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 68.6%

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

   if 8.4000000000000003e-4 < eps < 6.2000000000000001e47

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 83.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 83.8%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 83.8%

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

   if 6.2000000000000001e47 < eps < 1.99999999999999985e126

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 43.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 43.5%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 43.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 52.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 52.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \frac{{1}^{3} + {\varepsilon}^{3}}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1 \cdot 1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \frac{1}{4}\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon + 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left(1 + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{3}\right)\right)\right), \left(1 \cdot \color{blue}{1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 + \left(\color{blue}{\varepsilon \cdot \varepsilon} - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

       19. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 91.9%

       \[\leadsto 1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \color{blue}{\frac{\left(x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon - 1\right)}}\right) \]

   if 1.99999999999999985e126 < eps < 1.5499999999999999e184

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 81.2%

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

   if 1.5499999999999999e184 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 45.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 45.9%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 45.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 87.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 87.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 91.2%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00084:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{elif}\;\varepsilon \leq 6.2 \cdot 10^{+47}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+126}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + \frac{\left(x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{0 - x}\\ t_1 := x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\\ \mathbf{if}\;eps\_m \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\left(x + 1\right) \cdot t\_0\\ \mathbf{elif}\;eps\_m \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;eps\_m \leq 3.1 \cdot 10^{+125}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(eps\_m + 1\right) + \frac{t\_1 \cdot \left(1 + eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)}{1 + eps\_m \cdot \left(eps\_m + -1\right)}\right)\\ \mathbf{elif}\;eps\_m \leq 1.45 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- 0.0 x))) (t_1 (* x (* (+ eps_m 1.0) 0.25))))
   (if (<= eps_m 2.9e-22)
     (* (+ x 1.0) t_0)
     (if (<= eps_m 5.2e+47)
       t_0
       (if (<= eps_m 3.1e+125)
         (+
          1.0
          (*
           x
           (+
            (* -0.5 (+ eps_m 1.0))
            (/
             (* t_1 (+ 1.0 (* eps_m (* eps_m eps_m))))
             (+ 1.0 (* eps_m (+ eps_m -1.0)))))))
         (if (<= eps_m 1.45e+184)
           (+
            1.0
            (*
             x
             (*
              0.5
              (+
               (+ eps_m (- -1.0 eps_m))
               (*
                x
                (*
                 0.5
                 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
           (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 t_1))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((0.0 - x));
	double t_1 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 2.9e-22) {
		tmp = (x + 1.0) * t_0;
	} else if (eps_m <= 5.2e+47) {
		tmp = t_0;
	} else if (eps_m <= 3.1e+125) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_1 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 1.45e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_1)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((0.0d0 - x))
    t_1 = x * ((eps_m + 1.0d0) * 0.25d0)
    if (eps_m <= 2.9d-22) then
        tmp = (x + 1.0d0) * t_0
    else if (eps_m <= 5.2d+47) then
        tmp = t_0
    else if (eps_m <= 3.1d+125) then
        tmp = 1.0d0 + (x * (((-0.5d0) * (eps_m + 1.0d0)) + ((t_1 * (1.0d0 + (eps_m * (eps_m * eps_m)))) / (1.0d0 + (eps_m * (eps_m + (-1.0d0)))))))
    else if (eps_m <= 1.45d+184) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))
    else
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + t_1)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((0.0 - x));
	double t_1 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 2.9e-22) {
		tmp = (x + 1.0) * t_0;
	} else if (eps_m <= 5.2e+47) {
		tmp = t_0;
	} else if (eps_m <= 3.1e+125) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_1 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 1.45e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_1)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((0.0 - x))
	t_1 = x * ((eps_m + 1.0) * 0.25)
	tmp = 0
	if eps_m <= 2.9e-22:
		tmp = (x + 1.0) * t_0
	elif eps_m <= 5.2e+47:
		tmp = t_0
	elif eps_m <= 3.1e+125:
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_1 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))))
	elif eps_m <= 1.45e+184:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))))
	else:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_1)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(0.0 - x))
	t_1 = Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))
	tmp = 0.0
	if (eps_m <= 2.9e-22)
		tmp = Float64(Float64(x + 1.0) * t_0);
	elseif (eps_m <= 5.2e+47)
		tmp = t_0;
	elseif (eps_m <= 3.1e+125)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-0.5 * Float64(eps_m + 1.0)) + Float64(Float64(t_1 * Float64(1.0 + Float64(eps_m * Float64(eps_m * eps_m)))) / Float64(1.0 + Float64(eps_m * Float64(eps_m + -1.0)))))));
	elseif (eps_m <= 1.45e+184)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + t_1))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((0.0 - x));
	t_1 = x * ((eps_m + 1.0) * 0.25);
	tmp = 0.0;
	if (eps_m <= 2.9e-22)
		tmp = (x + 1.0) * t_0;
	elseif (eps_m <= 5.2e+47)
		tmp = t_0;
	elseif (eps_m <= 3.1e+125)
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_1 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	elseif (eps_m <= 1.45e+184)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	else
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 2.9e-22], N[(N[(x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[eps$95$m, 5.2e+47], t$95$0, If[LessEqual[eps$95$m, 3.1e+125], N[(1.0 + N[(x * N[(N[(-0.5 * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(1.0 + N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.45e+184], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < 2.9000000000000002e-22

   1. Initial program 61.1%

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

   3. Simplified 61.1%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

      3. distribute-rgt-out-- N/A

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x + 1\right), \color{blue}{\left(e^{-1 \cdot x}\right)}\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{\color{blue}{-1 \cdot x}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right) \]

      12. --lowering--.f64 67.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 67.9%

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

   if 2.9000000000000002e-22 < eps < 5.20000000000000007e47

   1. Initial program 96.4%

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

   3. Simplified 96.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{e^{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      4. --lowering--.f64 87.4%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   10. Simplified 87.4%

       \[\leadsto \color{blue}{e^{0 - x}} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

       2. neg-lowering-neg.f64 87.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right) \]

   12. Applied egg-rr 87.4%

       \[\leadsto e^{\color{blue}{-x}} \]

   if 5.20000000000000007e47 < eps < 3.1e125

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 43.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 43.5%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 43.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 52.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 52.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \frac{{1}^{3} + {\varepsilon}^{3}}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1 \cdot 1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \frac{1}{4}\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon + 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left(1 + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{3}\right)\right)\right), \left(1 \cdot \color{blue}{1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 + \left(\color{blue}{\varepsilon \cdot \varepsilon} - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

       19. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 91.9%

       \[\leadsto 1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \color{blue}{\frac{\left(x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon - 1\right)}}\right) \]

   if 3.1e125 < eps < 1.4499999999999999e184

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 81.2%

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

   if 1.4499999999999999e184 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 45.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 45.9%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 45.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 87.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 87.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 91.2%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.9 \cdot 10^{-22}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{elif}\;\varepsilon \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;e^{0 - x}\\ \mathbf{elif}\;\varepsilon \leq 3.1 \cdot 10^{+125}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + \frac{\left(x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{elif}\;\varepsilon \leq 1.45 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\\ \mathbf{if}\;eps\_m \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;e^{0 - x}\\ \mathbf{elif}\;eps\_m \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(eps\_m + 1\right) + \frac{t\_0 \cdot \left(1 + eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)}{1 + eps\_m \cdot \left(eps\_m + -1\right)}\right)\\ \mathbf{elif}\;eps\_m \leq 6.8 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* (+ eps_m 1.0) 0.25))))
   (if (<= eps_m 5.2e+47)
     (exp (- 0.0 x))
     (if (<= eps_m 1.3e+126)
       (+
        1.0
        (*
         x
         (+
          (* -0.5 (+ eps_m 1.0))
          (/
           (* t_0 (+ 1.0 (* eps_m (* eps_m eps_m))))
           (+ 1.0 (* eps_m (+ eps_m -1.0)))))))
       (if (<= eps_m 6.8e+184)
         (+
          1.0
          (*
           x
           (*
            0.5
            (+
             (+ eps_m (- -1.0 eps_m))
             (*
              x
              (* 0.5 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
         (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 t_0)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 5.2e+47) {
		tmp = exp((0.0 - x));
	} else if (eps_m <= 1.3e+126) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 6.8e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((eps_m + 1.0d0) * 0.25d0)
    if (eps_m <= 5.2d+47) then
        tmp = exp((0.0d0 - x))
    else if (eps_m <= 1.3d+126) then
        tmp = 1.0d0 + (x * (((-0.5d0) * (eps_m + 1.0d0)) + ((t_0 * (1.0d0 + (eps_m * (eps_m * eps_m)))) / (1.0d0 + (eps_m * (eps_m + (-1.0d0)))))))
    else if (eps_m <= 6.8d+184) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))
    else
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + t_0)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (eps_m <= 5.2e+47) {
		tmp = Math.exp((0.0 - x));
	} else if (eps_m <= 1.3e+126) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (eps_m <= 6.8e+184) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * ((eps_m + 1.0) * 0.25)
	tmp = 0
	if eps_m <= 5.2e+47:
		tmp = math.exp((0.0 - x))
	elif eps_m <= 1.3e+126:
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))))
	elif eps_m <= 6.8e+184:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))))
	else:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))
	tmp = 0.0
	if (eps_m <= 5.2e+47)
		tmp = exp(Float64(0.0 - x));
	elseif (eps_m <= 1.3e+126)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-0.5 * Float64(eps_m + 1.0)) + Float64(Float64(t_0 * Float64(1.0 + Float64(eps_m * Float64(eps_m * eps_m)))) / Float64(1.0 + Float64(eps_m * Float64(eps_m + -1.0)))))));
	elseif (eps_m <= 6.8e+184)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + t_0))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * ((eps_m + 1.0) * 0.25);
	tmp = 0.0;
	if (eps_m <= 5.2e+47)
		tmp = exp((0.0 - x));
	elseif (eps_m <= 1.3e+126)
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	elseif (eps_m <= 6.8e+184)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	else
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 5.2e+47], N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision], If[LessEqual[eps$95$m, 1.3e+126], N[(1.0 + N[(x * N[(N[(-0.5 * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(1.0 + N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 6.8e+184], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 5.20000000000000007e47

   1. Initial program 65.7%

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

   3. Simplified 65.7%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 97.8%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 97.8%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{e^{-1 \cdot x}} \]
   9. Step-by-step derivation

      1. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(-1 \cdot x\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{exp.f64}\left(\left(0 - x\right)\right) \]

      4. --lowering--.f64 77.6%

         \[\leadsto \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right) \]

   10. Simplified 77.6%

       \[\leadsto \color{blue}{e^{0 - x}} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right) \]

       2. neg-lowering-neg.f64 77.6%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(x\right)\right) \]

   12. Applied egg-rr 77.6%

       \[\leadsto e^{\color{blue}{-x}} \]

   if 5.20000000000000007e47 < eps < 1.3e126

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 43.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 43.5%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 43.5%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 52.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 52.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \frac{{1}^{3} + {\varepsilon}^{3}}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1 \cdot 1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \frac{1}{4}\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon + 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left(1 + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{3}\right)\right)\right), \left(1 \cdot \color{blue}{1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 + \left(\color{blue}{\varepsilon \cdot \varepsilon} - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

       19. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 91.9%

       \[\leadsto 1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \color{blue}{\frac{\left(x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon - 1\right)}}\right) \]

   if 1.3e126 < eps < 6.8000000000000003e184

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 81.2%

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

   if 6.8000000000000003e184 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 45.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 45.9%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 45.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 87.1%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 87.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 91.2%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.2 \cdot 10^{+47}:\\ \;\;\;\;e^{0 - x}\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + \frac{\left(x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{elif}\;\varepsilon \leq 6.8 \cdot 10^{+184}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\\ t_1 := -0.5 \cdot \left(eps\_m + 1\right)\\ t_2 := x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-111}:\\ \;\;\;\;1 + x \cdot \left(t\_1 + x \cdot \left(-0.08333333333333333 \cdot \left(x \cdot \left(\left(eps\_m + 1\right) \cdot t\_0\right)\right) + 0.25 \cdot t\_0\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + t\_2\right)\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-41}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + t\_0\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 480000000:\\ \;\;\;\;1 + x \cdot \left(t\_1 + \frac{t\_2 \cdot \left(1 + eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)}{1 + eps\_m \cdot \left(eps\_m + -1\right)}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+171}:\\ \;\;\;\;\left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ eps_m 1.0) (+ eps_m 1.0)))
        (t_1 (* -0.5 (+ eps_m 1.0)))
        (t_2 (* x (* (+ eps_m 1.0) 0.25))))
   (if (<= x -5e-111)
     (+
      1.0
      (*
       x
       (+
        t_1
        (*
         x
         (+
          (* -0.08333333333333333 (* x (* (+ eps_m 1.0) t_0)))
          (* 0.25 t_0))))))
     (if (<= x 2.4e-209)
       (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 t_2))))
       (if (<= x 2.7e-41)
         (+
          1.0
          (*
           x
           (*
            0.5
            (+
             (+ eps_m (- -1.0 eps_m))
             (* x (* 0.5 (+ (* eps_m eps_m) t_0)))))))
         (if (<= x 480000000.0)
           (+
            1.0
            (*
             x
             (+
              t_1
              (/
               (* t_2 (+ 1.0 (* eps_m (* eps_m eps_m))))
               (+ 1.0 (* eps_m (+ eps_m -1.0)))))))
           (if (<= x 1.65e+171) (* (* 0.5 (* eps_m eps_m)) (* x x)) 0.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (eps_m + 1.0) * (eps_m + 1.0);
	double t_1 = -0.5 * (eps_m + 1.0);
	double t_2 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (x <= -5e-111) {
		tmp = 1.0 + (x * (t_1 + (x * ((-0.08333333333333333 * (x * ((eps_m + 1.0) * t_0))) + (0.25 * t_0)))));
	} else if (x <= 2.4e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_2)));
	} else if (x <= 2.7e-41) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))));
	} else if (x <= 480000000.0) {
		tmp = 1.0 + (x * (t_1 + ((t_2 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (x <= 1.65e+171) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (eps_m + 1.0d0) * (eps_m + 1.0d0)
    t_1 = (-0.5d0) * (eps_m + 1.0d0)
    t_2 = x * ((eps_m + 1.0d0) * 0.25d0)
    if (x <= (-5d-111)) then
        tmp = 1.0d0 + (x * (t_1 + (x * (((-0.08333333333333333d0) * (x * ((eps_m + 1.0d0) * t_0))) + (0.25d0 * t_0)))))
    else if (x <= 2.4d-209) then
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + t_2)))
    else if (x <= 2.7d-41) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + t_0))))))
    else if (x <= 480000000.0d0) then
        tmp = 1.0d0 + (x * (t_1 + ((t_2 * (1.0d0 + (eps_m * (eps_m * eps_m)))) / (1.0d0 + (eps_m * (eps_m + (-1.0d0)))))))
    else if (x <= 1.65d+171) then
        tmp = (0.5d0 * (eps_m * eps_m)) * (x * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (eps_m + 1.0) * (eps_m + 1.0);
	double t_1 = -0.5 * (eps_m + 1.0);
	double t_2 = x * ((eps_m + 1.0) * 0.25);
	double tmp;
	if (x <= -5e-111) {
		tmp = 1.0 + (x * (t_1 + (x * ((-0.08333333333333333 * (x * ((eps_m + 1.0) * t_0))) + (0.25 * t_0)))));
	} else if (x <= 2.4e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_2)));
	} else if (x <= 2.7e-41) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))));
	} else if (x <= 480000000.0) {
		tmp = 1.0 + (x * (t_1 + ((t_2 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (x <= 1.65e+171) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (eps_m + 1.0) * (eps_m + 1.0)
	t_1 = -0.5 * (eps_m + 1.0)
	t_2 = x * ((eps_m + 1.0) * 0.25)
	tmp = 0
	if x <= -5e-111:
		tmp = 1.0 + (x * (t_1 + (x * ((-0.08333333333333333 * (x * ((eps_m + 1.0) * t_0))) + (0.25 * t_0)))))
	elif x <= 2.4e-209:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_2)))
	elif x <= 2.7e-41:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))))
	elif x <= 480000000.0:
		tmp = 1.0 + (x * (t_1 + ((t_2 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))))
	elif x <= 1.65e+171:
		tmp = (0.5 * (eps_m * eps_m)) * (x * x)
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))
	t_1 = Float64(-0.5 * Float64(eps_m + 1.0))
	t_2 = Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))
	tmp = 0.0
	if (x <= -5e-111)
		tmp = Float64(1.0 + Float64(x * Float64(t_1 + Float64(x * Float64(Float64(-0.08333333333333333 * Float64(x * Float64(Float64(eps_m + 1.0) * t_0))) + Float64(0.25 * t_0))))));
	elseif (x <= 2.4e-209)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + t_2))));
	elseif (x <= 2.7e-41)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + t_0)))))));
	elseif (x <= 480000000.0)
		tmp = Float64(1.0 + Float64(x * Float64(t_1 + Float64(Float64(t_2 * Float64(1.0 + Float64(eps_m * Float64(eps_m * eps_m)))) / Float64(1.0 + Float64(eps_m * Float64(eps_m + -1.0)))))));
	elseif (x <= 1.65e+171)
		tmp = Float64(Float64(0.5 * Float64(eps_m * eps_m)) * Float64(x * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (eps_m + 1.0) * (eps_m + 1.0);
	t_1 = -0.5 * (eps_m + 1.0);
	t_2 = x * ((eps_m + 1.0) * 0.25);
	tmp = 0.0;
	if (x <= -5e-111)
		tmp = 1.0 + (x * (t_1 + (x * ((-0.08333333333333333 * (x * ((eps_m + 1.0) * t_0))) + (0.25 * t_0)))));
	elseif (x <= 2.4e-209)
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_2)));
	elseif (x <= 2.7e-41)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))));
	elseif (x <= 480000000.0)
		tmp = 1.0 + (x * (t_1 + ((t_2 * (1.0 + (eps_m * (eps_m * eps_m)))) / (1.0 + (eps_m * (eps_m + -1.0))))));
	elseif (x <= 1.65e+171)
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-111], N[(1.0 + N[(x * N[(t$95$1 + N[(x * N[(N[(-0.08333333333333333 * N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-209], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-41], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 480000000.0], N[(1.0 + N[(x * N[(t$95$1 + N[(N[(t$95$2 * N[(1.0 + N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+171], N[(N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -5.0000000000000003e-111

   1. Initial program 80.5%

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

   3. Simplified 80.5%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 46.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 46.7%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 59.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{x} \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(x \cdot \left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{12} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right), \color{blue}{\left(\frac{1}{4} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   13. Simplified 55.8%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + x \cdot \left(-0.08333333333333333 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right) + \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot 0.25\right)\right)} \]

   if -5.0000000000000003e-111 < x < 2.4000000000000001e-209

   1. Initial program 44.1%

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

   3. Simplified 44.1%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 37.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 37.1%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 92.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 83.9%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 83.9%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 94.8%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]

   if 2.4000000000000001e-209 < x < 2.7e-41

   1. Initial program 66.4%

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

   3. Simplified 66.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 98.0%

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

   if 2.7e-41 < x < 4.8e8

   1. Initial program 61.3%

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

   3. Simplified 61.3%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 22.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 22.0%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 48.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 30.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 30.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \frac{{1}^{3} + {\varepsilon}^{3}}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1 \cdot 1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \frac{1}{4}\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon + 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left(1 + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{3}\right)\right)\right), \left(1 \cdot \color{blue}{1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 + \left(\color{blue}{\varepsilon \cdot \varepsilon} - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

       19. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 87.2%

       \[\leadsto 1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \color{blue}{\frac{\left(x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon - 1\right)}}\right) \]

   if 4.8e8 < x < 1.64999999999999996e171

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 3.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 68.9%

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

   if 1.64999999999999996e171 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-111}:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + x \cdot \left(-0.08333333333333333 \cdot \left(x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 0.25 \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-41}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 480000000:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + \frac{\left(x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+171}:\\ \;\;\;\;\left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\\ t_1 := eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-54}:\\ \;\;\;\;1 + x \cdot \left(t\_1 \cdot \frac{x \cdot \left(0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{eps\_m}\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + t\_0\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-41}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 480000000:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(eps\_m + 1\right) + \frac{t\_0 \cdot \left(1 + t\_1\right)}{1 + eps\_m \cdot \left(eps\_m + -1\right)}\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+167}:\\ \;\;\;\;\left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* (+ eps_m 1.0) 0.25))) (t_1 (* eps_m (* eps_m eps_m))))
   (if (<= x -1.95e-54)
     (+
      1.0
      (*
       x
       (*
        t_1
        (/
         (*
          x
          (+ 0.25 (+ (* x -0.25) (* 0.5 (+ 0.5 (* x -0.16666666666666666))))))
         eps_m))))
     (if (<= x 3.6e-209)
       (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 t_0))))
       (if (<= x 2.9e-41)
         (+
          1.0
          (*
           x
           (*
            0.5
            (+
             (+ eps_m (- -1.0 eps_m))
             (*
              x
              (* 0.5 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
         (if (<= x 480000000.0)
           (+
            1.0
            (*
             x
             (+
              (* -0.5 (+ eps_m 1.0))
              (/ (* t_0 (+ 1.0 t_1)) (+ 1.0 (* eps_m (+ eps_m -1.0)))))))
           (if (<= x 1.7e+167) (* (* 0.5 (* eps_m eps_m)) (* x x)) 0.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double t_1 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -1.95e-54) {
		tmp = 1.0 + (x * (t_1 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)));
	} else if (x <= 3.6e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	} else if (x <= 2.9e-41) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else if (x <= 480000000.0) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + t_1)) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (x <= 1.7e+167) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * ((eps_m + 1.0d0) * 0.25d0)
    t_1 = eps_m * (eps_m * eps_m)
    if (x <= (-1.95d-54)) then
        tmp = 1.0d0 + (x * (t_1 * ((x * (0.25d0 + ((x * (-0.25d0)) + (0.5d0 * (0.5d0 + (x * (-0.16666666666666666d0))))))) / eps_m)))
    else if (x <= 3.6d-209) then
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + t_0)))
    else if (x <= 2.9d-41) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))
    else if (x <= 480000000.0d0) then
        tmp = 1.0d0 + (x * (((-0.5d0) * (eps_m + 1.0d0)) + ((t_0 * (1.0d0 + t_1)) / (1.0d0 + (eps_m * (eps_m + (-1.0d0)))))))
    else if (x <= 1.7d+167) then
        tmp = (0.5d0 * (eps_m * eps_m)) * (x * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * ((eps_m + 1.0) * 0.25);
	double t_1 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -1.95e-54) {
		tmp = 1.0 + (x * (t_1 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)));
	} else if (x <= 3.6e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	} else if (x <= 2.9e-41) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else if (x <= 480000000.0) {
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + t_1)) / (1.0 + (eps_m * (eps_m + -1.0))))));
	} else if (x <= 1.7e+167) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * ((eps_m + 1.0) * 0.25)
	t_1 = eps_m * (eps_m * eps_m)
	tmp = 0
	if x <= -1.95e-54:
		tmp = 1.0 + (x * (t_1 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)))
	elif x <= 3.6e-209:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)))
	elif x <= 2.9e-41:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))))
	elif x <= 480000000.0:
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + t_1)) / (1.0 + (eps_m * (eps_m + -1.0))))))
	elif x <= 1.7e+167:
		tmp = (0.5 * (eps_m * eps_m)) * (x * x)
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))
	t_1 = Float64(eps_m * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= -1.95e-54)
		tmp = Float64(1.0 + Float64(x * Float64(t_1 * Float64(Float64(x * Float64(0.25 + Float64(Float64(x * -0.25) + Float64(0.5 * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / eps_m))));
	elseif (x <= 3.6e-209)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + t_0))));
	elseif (x <= 2.9e-41)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))));
	elseif (x <= 480000000.0)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-0.5 * Float64(eps_m + 1.0)) + Float64(Float64(t_0 * Float64(1.0 + t_1)) / Float64(1.0 + Float64(eps_m * Float64(eps_m + -1.0)))))));
	elseif (x <= 1.7e+167)
		tmp = Float64(Float64(0.5 * Float64(eps_m * eps_m)) * Float64(x * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * ((eps_m + 1.0) * 0.25);
	t_1 = eps_m * (eps_m * eps_m);
	tmp = 0.0;
	if (x <= -1.95e-54)
		tmp = 1.0 + (x * (t_1 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)));
	elseif (x <= 3.6e-209)
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + t_0)));
	elseif (x <= 2.9e-41)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	elseif (x <= 480000000.0)
		tmp = 1.0 + (x * ((-0.5 * (eps_m + 1.0)) + ((t_0 * (1.0 + t_1)) / (1.0 + (eps_m * (eps_m + -1.0))))));
	elseif (x <= 1.7e+167)
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-54], N[(1.0 + N[(x * N[(t$95$1 * N[(N[(x * N[(0.25 + N[(N[(x * -0.25), $MachinePrecision] + N[(0.5 * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-209], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-41], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 480000000.0], N[(1.0 + N[(x * N[(N[(-0.5 * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.7e+167], N[(N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.95e-54

   1. Initial program 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 21.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({\varepsilon}^{3} \cdot \left(x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + \color{blue}{x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + x \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + x \cdot \left(x \cdot 0\right)\right)\right)\right)\right) \]

      10. mul0-rgt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + x \cdot 0\right)\right)\right)\right) \]

      11. mul0-rgt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + 0\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right), \color{blue}{0}\right)\right)\right)\right) \]

   10. Simplified 81.4%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\frac{x \cdot \left(0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{\varepsilon} + 0\right)\right)} \]

   if -1.95e-54 < x < 3.60000000000000016e-209

   1. Initial program 46.1%

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

   3. Simplified 46.1%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 38.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 38.0%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 91.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 85.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 85.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 94.6%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]

   if 3.60000000000000016e-209 < x < 2.89999999999999977e-41

   1. Initial program 66.4%

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

   3. Simplified 66.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 98.0%

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

   if 2.89999999999999977e-41 < x < 4.8e8

   1. Initial program 61.3%

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

   3. Simplified 61.3%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 22.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 22.0%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 48.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 30.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 30.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \frac{{1}^{3} + {\varepsilon}^{3}}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)}{\color{blue}{1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left({1}^{3} + {\varepsilon}^{3}\right)\right), \color{blue}{\left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot x\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1 \cdot 1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot \frac{1}{4}\right) \cdot \left(1 + \varepsilon\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(\color{blue}{1} \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(1 + \varepsilon\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       10. +-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \left(\varepsilon + 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left({1}^{3} + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \left(1 + {\varepsilon}^{3}\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left({\varepsilon}^{3}\right)\right)\right), \left(1 \cdot \color{blue}{1} + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 \cdot 1 + \left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       17. metadata-eval N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \left(1 + \left(\color{blue}{\varepsilon \cdot \varepsilon} - 1 \cdot \varepsilon\right)\right)\right)\right)\right)\right) \]

       18. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\varepsilon \cdot \varepsilon - 1 \cdot \varepsilon\right)}\right)\right)\right)\right)\right) \]

       19. distribute-rgt-out-- N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\varepsilon \cdot \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\varepsilon, 1\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon - 1\right)}\right)\right)\right)\right)\right)\right) \]

   15. Applied egg-rr 87.2%

       \[\leadsto 1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \color{blue}{\frac{\left(x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon - 1\right)}}\right) \]

   if 4.8e8 < x < 1.7e167

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 3.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 68.9%

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

   if 1.7e167 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-54}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{x \cdot \left(0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-41}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 480000000:\\ \;\;\;\;1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + 1\right) + \frac{\left(x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right) \cdot \left(1 + \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{1 + \varepsilon \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+167}:\\ \;\;\;\;\left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-55}:\\ \;\;\;\;1 + x \cdot \left(t\_0 \cdot \frac{x \cdot \left(0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{eps\_m}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+166}:\\ \;\;\;\;\left(t\_0 \cdot 0.08333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* eps_m (* eps_m eps_m))))
   (if (<= x -9.8e-55)
     (+
      1.0
      (*
       x
       (*
        t_0
        (/
         (*
          x
          (+ 0.25 (+ (* x -0.25) (* 0.5 (+ 0.5 (* x -0.16666666666666666))))))
         eps_m))))
     (if (<= x 3.5e-212)
       (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 (* x (* (+ eps_m 1.0) 0.25))))))
       (if (<= x 3.1e-33)
         (+
          1.0
          (*
           x
           (*
            0.5
            (+
             (+ eps_m (- -1.0 eps_m))
             (*
              x
              (* 0.5 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
         (if (<= x 5.5e+166)
           (* (* t_0 0.08333333333333333) (* x (* x x)))
           0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -9.8e-55) {
		tmp = 1.0 + (x * (t_0 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)));
	} else if (x <= 3.5e-212) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else if (x <= 5.5e+166) {
		tmp = (t_0 * 0.08333333333333333) * (x * (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = eps_m * (eps_m * eps_m)
    if (x <= (-9.8d-55)) then
        tmp = 1.0d0 + (x * (t_0 * ((x * (0.25d0 + ((x * (-0.25d0)) + (0.5d0 * (0.5d0 + (x * (-0.16666666666666666d0))))))) / eps_m)))
    else if (x <= 3.5d-212) then
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + (x * ((eps_m + 1.0d0) * 0.25d0)))))
    else if (x <= 3.1d-33) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))
    else if (x <= 5.5d+166) then
        tmp = (t_0 * 0.08333333333333333d0) * (x * (x * x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -9.8e-55) {
		tmp = 1.0 + (x * (t_0 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)));
	} else if (x <= 3.5e-212) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	} else if (x <= 5.5e+166) {
		tmp = (t_0 * 0.08333333333333333) * (x * (x * x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = eps_m * (eps_m * eps_m)
	tmp = 0
	if x <= -9.8e-55:
		tmp = 1.0 + (x * (t_0 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)))
	elif x <= 3.5e-212:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))))
	elif x <= 3.1e-33:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))))
	elif x <= 5.5e+166:
		tmp = (t_0 * 0.08333333333333333) * (x * (x * x))
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(eps_m * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= -9.8e-55)
		tmp = Float64(1.0 + Float64(x * Float64(t_0 * Float64(Float64(x * Float64(0.25 + Float64(Float64(x * -0.25) + Float64(0.5 * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / eps_m))));
	elseif (x <= 3.5e-212)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))))));
	elseif (x <= 3.1e-33)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))));
	elseif (x <= 5.5e+166)
		tmp = Float64(Float64(t_0 * 0.08333333333333333) * Float64(x * Float64(x * x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = eps_m * (eps_m * eps_m);
	tmp = 0.0;
	if (x <= -9.8e-55)
		tmp = 1.0 + (x * (t_0 * ((x * (0.25 + ((x * -0.25) + (0.5 * (0.5 + (x * -0.16666666666666666)))))) / eps_m)));
	elseif (x <= 3.5e-212)
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	elseif (x <= 3.1e-33)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0))))))));
	elseif (x <= 5.5e+166)
		tmp = (t_0 * 0.08333333333333333) * (x * (x * x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-55], N[(1.0 + N[(x * N[(t$95$0 * N[(N[(x * N[(0.25 + N[(N[(x * -0.25), $MachinePrecision] + N[(0.5 * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e-212], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-33], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+166], N[(N[(t$95$0 * 0.08333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.80000000000000071e-55

   1. Initial program 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 21.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({\varepsilon}^{3} \cdot \left(x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + \color{blue}{x \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)}\right)\right)\right)\right) \]

      8. distribute-rgt-out N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + x \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + x \cdot \left(x \cdot 0\right)\right)\right)\right)\right) \]

      10. mul0-rgt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + x \cdot 0\right)\right)\right)\right) \]

      11. mul0-rgt N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon} + 0\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(\frac{x \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right), \color{blue}{0}\right)\right)\right)\right) \]

   10. Simplified 81.4%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\frac{x \cdot \left(0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{\varepsilon} + 0\right)\right)} \]

   if -9.80000000000000071e-55 < x < 3.4999999999999998e-212

   1. Initial program 46.1%

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

   3. Simplified 46.1%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 38.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 38.0%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 91.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 85.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 85.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 94.6%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]

   if 3.4999999999999998e-212 < x < 3.09999999999999997e-33

   1. Initial program 64.6%

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

   3. Simplified 64.6%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 96.3%

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

   if 3.09999999999999997e-33 < x < 5.50000000000000008e166

   1. Initial program 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(-1, \varepsilon\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(-1 + \varepsilon\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + -1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon - 1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + -1\right) + x \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right)\right)\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      13. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 43.8%

       \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 5.50000000000000008e166 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-55}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \frac{x \cdot \left(0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+166}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.08333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\\ t_1 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -0.16:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(-0.5 + \frac{0.5}{eps\_m}\right) \cdot \left(\left(\left(eps\_m + 1\right) \cdot t\_0\right) \cdot t\_1\right)\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + t\_0\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+170}:\\ \;\;\;\;\left(\left(eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot 0.08333333333333333\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (+ eps_m 1.0) (+ eps_m 1.0))) (t_1 (* x (* x x))))
   (if (<= x -0.16)
     (*
      0.16666666666666666
      (* (+ -0.5 (/ 0.5 eps_m)) (* (* (+ eps_m 1.0) t_0) t_1)))
     (if (<= x 6.5e-209)
       (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 (* x (* (+ eps_m 1.0) 0.25))))))
       (if (<= x 3.1e-33)
         (+
          1.0
          (*
           x
           (*
            0.5
            (+
             (+ eps_m (- -1.0 eps_m))
             (* x (* 0.5 (+ (* eps_m eps_m) t_0)))))))
         (if (<= x 9.4e+170)
           (* (* (* eps_m (* eps_m eps_m)) 0.08333333333333333) t_1)
           0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (eps_m + 1.0) * (eps_m + 1.0);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= -0.16) {
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * t_0) * t_1));
	} else if (x <= 6.5e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))));
	} else if (x <= 9.4e+170) {
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (eps_m + 1.0d0) * (eps_m + 1.0d0)
    t_1 = x * (x * x)
    if (x <= (-0.16d0)) then
        tmp = 0.16666666666666666d0 * (((-0.5d0) + (0.5d0 / eps_m)) * (((eps_m + 1.0d0) * t_0) * t_1))
    else if (x <= 6.5d-209) then
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + (x * ((eps_m + 1.0d0) * 0.25d0)))))
    else if (x <= 3.1d-33) then
        tmp = 1.0d0 + (x * (0.5d0 * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + t_0))))))
    else if (x <= 9.4d+170) then
        tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333d0) * t_1
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (eps_m + 1.0) * (eps_m + 1.0);
	double t_1 = x * (x * x);
	double tmp;
	if (x <= -0.16) {
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * t_0) * t_1));
	} else if (x <= 6.5e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))));
	} else if (x <= 9.4e+170) {
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_1;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (eps_m + 1.0) * (eps_m + 1.0)
	t_1 = x * (x * x)
	tmp = 0
	if x <= -0.16:
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * t_0) * t_1))
	elif x <= 6.5e-209:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))))
	elif x <= 3.1e-33:
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))))
	elif x <= 9.4e+170:
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_1
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))
	t_1 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -0.16)
		tmp = Float64(0.16666666666666666 * Float64(Float64(-0.5 + Float64(0.5 / eps_m)) * Float64(Float64(Float64(eps_m + 1.0) * t_0) * t_1)));
	elseif (x <= 6.5e-209)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))))));
	elseif (x <= 3.1e-33)
		tmp = Float64(1.0 + Float64(x * Float64(0.5 * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + t_0)))))));
	elseif (x <= 9.4e+170)
		tmp = Float64(Float64(Float64(eps_m * Float64(eps_m * eps_m)) * 0.08333333333333333) * t_1);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (eps_m + 1.0) * (eps_m + 1.0);
	t_1 = x * (x * x);
	tmp = 0.0;
	if (x <= -0.16)
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * t_0) * t_1));
	elseif (x <= 6.5e-209)
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	elseif (x <= 3.1e-33)
		tmp = 1.0 + (x * (0.5 * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + t_0))))));
	elseif (x <= 9.4e+170)
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_1;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.16], N[(0.16666666666666666 * N[(N[(-0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-209], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-33], N[(1.0 + N[(x * N[(0.5 * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.4e+170], N[(N[(N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * t$95$1), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -0.160000000000000003

   1. Initial program 94.7%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 48.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 48.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)}, \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\varepsilon}\right), \frac{-1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   10. Simplified 39.8%

       \[\leadsto \left(0.5 + \frac{0.5}{\varepsilon}\right) - \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot \left(0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(x \cdot -0.16666666666666666\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{3} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\left({x}^{3} \cdot {\left(1 + \varepsilon\right)}^{3}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{3} \cdot {\left(1 + \varepsilon\right)}^{3}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({\left(1 + \varepsilon\right)}^{3} \cdot {x}^{3}\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(1 + \varepsilon\right)}^{3}\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \varepsilon\right) \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       19. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)} \]

   if -0.160000000000000003 < x < 6.50000000000000042e-209

   1. Initial program 48.9%

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

   3. Simplified 48.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 39.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 39.0%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 88.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 81.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 81.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 89.2%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]

   if 6.50000000000000042e-209 < x < 3.09999999999999997e-33

   1. Initial program 64.6%

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

   3. Simplified 64.6%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(e^{x \cdot \left(\varepsilon - 1\right)} + e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(e^{x \cdot \left(\varepsilon - 1\right)}\right), \color{blue}{\left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}\right)\right) \]

      6. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\color{blue}{-1} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \left(e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 \cdot 1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right) \]

      19. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right)\right)\right)\right) \]

      20. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\varepsilon, -1\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right)\right) \]

   10. Simplified 100.0%

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

   11. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. distribute-lft-out N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \color{blue}{\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       5. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \left(\varepsilon + \color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left(\varepsilon + \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right)\right)\right) \]

       7. associate-+r+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right) + \color{blue}{x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{+.f64}\left(\left(\varepsilon + -1 \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(x \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2} + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right)\right) \]

   13. Simplified 96.3%

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

   if 3.09999999999999997e-33 < x < 9.40000000000000008e170

   1. Initial program 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(-1, \varepsilon\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(-1 + \varepsilon\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + -1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon - 1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + -1\right) + x \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right)\right)\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      13. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 43.8%

       \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 9.40000000000000008e170 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.16:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(-0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + 1\right) \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(0.5 \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+170}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.08333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.9% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(-0.5 + \frac{0.5}{eps\_m}\right) \cdot \left(\left(\left(eps\_m + 1\right) \cdot \left(\left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right) \cdot t\_0\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-216}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot 0.08333333333333333\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= x -0.48)
     (*
      0.16666666666666666
      (*
       (+ -0.5 (/ 0.5 eps_m))
       (* (* (+ eps_m 1.0) (* (+ eps_m 1.0) (+ eps_m 1.0))) t_0)))
     (if (<= x 2e-216)
       (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 (* x (* (+ eps_m 1.0) 0.25))))))
       (if (<= x 3.1e-33)
         (+ 1.0 (* x (* x (* 0.25 (* eps_m eps_m)))))
         (if (<= x 3.4e+169)
           (* (* (* eps_m (* eps_m eps_m)) 0.08333333333333333) t_0)
           0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -0.48) {
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * ((eps_m + 1.0) * (eps_m + 1.0))) * t_0));
	} else if (x <= 2e-216) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 3.4e+169) {
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * x)
    if (x <= (-0.48d0)) then
        tmp = 0.16666666666666666d0 * (((-0.5d0) + (0.5d0 / eps_m)) * (((eps_m + 1.0d0) * ((eps_m + 1.0d0) * (eps_m + 1.0d0))) * t_0))
    else if (x <= 2d-216) then
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + (x * ((eps_m + 1.0d0) * 0.25d0)))))
    else if (x <= 3.1d-33) then
        tmp = 1.0d0 + (x * (x * (0.25d0 * (eps_m * eps_m))))
    else if (x <= 3.4d+169) then
        tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333d0) * t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (x * x);
	double tmp;
	if (x <= -0.48) {
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * ((eps_m + 1.0) * (eps_m + 1.0))) * t_0));
	} else if (x <= 2e-216) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 3.4e+169) {
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (x * x)
	tmp = 0
	if x <= -0.48:
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * ((eps_m + 1.0) * (eps_m + 1.0))) * t_0))
	elif x <= 2e-216:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))))
	elif x <= 3.1e-33:
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))))
	elif x <= 3.4e+169:
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (x <= -0.48)
		tmp = Float64(0.16666666666666666 * Float64(Float64(-0.5 + Float64(0.5 / eps_m)) * Float64(Float64(Float64(eps_m + 1.0) * Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))) * t_0)));
	elseif (x <= 2e-216)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))))));
	elseif (x <= 3.1e-33)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.25 * Float64(eps_m * eps_m)))));
	elseif (x <= 3.4e+169)
		tmp = Float64(Float64(Float64(eps_m * Float64(eps_m * eps_m)) * 0.08333333333333333) * t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (x * x);
	tmp = 0.0;
	if (x <= -0.48)
		tmp = 0.16666666666666666 * ((-0.5 + (0.5 / eps_m)) * (((eps_m + 1.0) * ((eps_m + 1.0) * (eps_m + 1.0))) * t_0));
	elseif (x <= 2e-216)
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	elseif (x <= 3.1e-33)
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	elseif (x <= 3.4e+169)
		tmp = ((eps_m * (eps_m * eps_m)) * 0.08333333333333333) * t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.48], N[(0.16666666666666666 * N[(N[(-0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e-216], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-33], N[(1.0 + N[(x * N[(x * N[(0.25 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e+169], N[(N[(N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * t$95$0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -0.47999999999999998

   1. Initial program 94.7%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 48.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 48.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)}, \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\varepsilon}\right), \frac{-1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   10. Simplified 39.8%

       \[\leadsto \left(0.5 + \frac{0.5}{\varepsilon}\right) - \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot \left(0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(x \cdot -0.16666666666666666\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({x}^{3} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{3} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(\left({x}^{3} \cdot {\left(1 + \varepsilon\right)}^{3}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({x}^{3} \cdot {\left(1 + \varepsilon\right)}^{3}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({\left(1 + \varepsilon\right)}^{3} \cdot {x}^{3}\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(1 + \varepsilon\right)}^{3}\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2} \cdot \frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       6. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(1 + \varepsilon\right) \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left({x}^{3}\right)\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) \]

       14. cube-mult N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(x \cdot {x}^{2}\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\color{blue}{\varepsilon}} - \frac{1}{2}\right)\right)\right) \]

       19. sub-neg N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

       20. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} + \frac{-1}{2}\right)\right)\right) \]

       21. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right), \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)} \]

   if -0.47999999999999998 < x < 2.0000000000000001e-216

   1. Initial program 48.9%

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

   3. Simplified 48.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 39.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 39.0%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 88.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 81.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 81.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 89.2%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]

   if 2.0000000000000001e-216 < x < 3.09999999999999997e-33

   1. Initial program 64.6%

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

   3. Simplified 64.6%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 47.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 47.8%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 83.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 96.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 96.3%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 96.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   16. Simplified 96.3%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   if 3.09999999999999997e-33 < x < 3.40000000000000028e169

   1. Initial program 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(-1, \varepsilon\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(-1 + \varepsilon\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + -1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon - 1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + -1\right) + x \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right)\right)\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      13. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 43.8%

       \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 3.40000000000000028e169 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;0.16666666666666666 \cdot \left(\left(-0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + 1\right) \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-216}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.08333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{-27}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(eps\_m + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+170}:\\ \;\;\;\;\left(t\_1 \cdot 0.08333333333333333\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* eps_m (* eps_m eps_m))))
   (if (<= x -1.16e-27)
     (* t_0 (* t_1 -0.08333333333333333))
     (if (<= x 3.9e-209)
       (+ 1.0 (* x (* (+ eps_m 1.0) (+ -0.5 (* x (* (+ eps_m 1.0) 0.25))))))
       (if (<= x 3.1e-33)
         (+ 1.0 (* x (* x (* 0.25 (* eps_m eps_m)))))
         (if (<= x 9.4e+170) (* (* t_1 0.08333333333333333) t_0) 0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x * x);
	double t_1 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -1.16e-27) {
		tmp = t_0 * (t_1 * -0.08333333333333333);
	} else if (x <= 3.9e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 9.4e+170) {
		tmp = (t_1 * 0.08333333333333333) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = eps_m * (eps_m * eps_m)
    if (x <= (-1.16d-27)) then
        tmp = t_0 * (t_1 * (-0.08333333333333333d0))
    else if (x <= 3.9d-209) then
        tmp = 1.0d0 + (x * ((eps_m + 1.0d0) * ((-0.5d0) + (x * ((eps_m + 1.0d0) * 0.25d0)))))
    else if (x <= 3.1d-33) then
        tmp = 1.0d0 + (x * (x * (0.25d0 * (eps_m * eps_m))))
    else if (x <= 9.4d+170) then
        tmp = (t_1 * 0.08333333333333333d0) * t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (x * x);
	double t_1 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -1.16e-27) {
		tmp = t_0 * (t_1 * -0.08333333333333333);
	} else if (x <= 3.9e-209) {
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 9.4e+170) {
		tmp = (t_1 * 0.08333333333333333) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (x * x)
	t_1 = eps_m * (eps_m * eps_m)
	tmp = 0
	if x <= -1.16e-27:
		tmp = t_0 * (t_1 * -0.08333333333333333)
	elif x <= 3.9e-209:
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))))
	elif x <= 3.1e-33:
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))))
	elif x <= 9.4e+170:
		tmp = (t_1 * 0.08333333333333333) * t_0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(eps_m * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= -1.16e-27)
		tmp = Float64(t_0 * Float64(t_1 * -0.08333333333333333));
	elseif (x <= 3.9e-209)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-0.5 + Float64(x * Float64(Float64(eps_m + 1.0) * 0.25))))));
	elseif (x <= 3.1e-33)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.25 * Float64(eps_m * eps_m)))));
	elseif (x <= 9.4e+170)
		tmp = Float64(Float64(t_1 * 0.08333333333333333) * t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (x * x);
	t_1 = eps_m * (eps_m * eps_m);
	tmp = 0.0;
	if (x <= -1.16e-27)
		tmp = t_0 * (t_1 * -0.08333333333333333);
	elseif (x <= 3.9e-209)
		tmp = 1.0 + (x * ((eps_m + 1.0) * (-0.5 + (x * ((eps_m + 1.0) * 0.25)))));
	elseif (x <= 3.1e-33)
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	elseif (x <= 9.4e+170)
		tmp = (t_1 * 0.08333333333333333) * t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.16e-27], N[(t$95$0 * N[(t$95$1 * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.9e-209], N[(1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-0.5 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-33], N[(1.0 + N[(x * N[(x * N[(0.25 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.4e+170], N[(N[(t$95$1 * 0.08333333333333333), $MachinePrecision] * t$95$0), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.16000000000000005e-27

   1. Initial program 89.1%

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

   3. Simplified 89.1%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 48.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 48.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)}, \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\varepsilon}\right), \frac{-1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   10. Simplified 40.7%

       \[\leadsto \left(0.5 + \frac{0.5}{\varepsilon}\right) - \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot \left(0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(x \cdot -0.16666666666666666\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       13. *-lowering-*.f64 40.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 40.5%

       \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -1.16000000000000005e-27 < x < 3.9e-209

   1. Initial program 48.1%

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

   3. Simplified 48.1%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 38.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 38.4%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 90.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 84.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 84.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]
   14. Step-by-step derivation

       1. +-commutative N/A

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

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2} + \left(\frac{1}{4} \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right), \color{blue}{1}\right) \]

   15. Applied egg-rr 92.8%

       \[\leadsto \color{blue}{x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon + 1\right)\right)\right)\right) + 1} \]

   if 3.9e-209 < x < 3.09999999999999997e-33

   1. Initial program 64.6%

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

   3. Simplified 64.6%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 47.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 47.8%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 83.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 96.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 96.3%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 96.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   16. Simplified 96.3%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   if 3.09999999999999997e-33 < x < 9.40000000000000008e170

   1. Initial program 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(-1, \varepsilon\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(-1 + \varepsilon\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + -1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon - 1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + -1\right) + x \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right)\right)\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      13. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 43.8%

       \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 9.40000000000000008e170 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-27}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-209}:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-0.5 + x \cdot \left(\left(\varepsilon + 1\right) \cdot 0.25\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+170}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.08333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.9% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq -0.41:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+169}:\\ \;\;\;\;\left(t\_1 \cdot 0.08333333333333333\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* x x))) (t_1 (* eps_m (* eps_m eps_m))))
   (if (<= x -0.41)
     (* t_0 (* t_1 -0.08333333333333333))
     (if (<= x 3.1e-33)
       (+ 1.0 (* x (* x (* 0.25 (* eps_m eps_m)))))
       (if (<= x 7.4e+169) (* (* t_1 0.08333333333333333) t_0) 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x * x);
	double t_1 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -0.41) {
		tmp = t_0 * (t_1 * -0.08333333333333333);
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 7.4e+169) {
		tmp = (t_1 * 0.08333333333333333) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (x * x)
    t_1 = eps_m * (eps_m * eps_m)
    if (x <= (-0.41d0)) then
        tmp = t_0 * (t_1 * (-0.08333333333333333d0))
    else if (x <= 3.1d-33) then
        tmp = 1.0d0 + (x * (x * (0.25d0 * (eps_m * eps_m))))
    else if (x <= 7.4d+169) then
        tmp = (t_1 * 0.08333333333333333d0) * t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (x * x);
	double t_1 = eps_m * (eps_m * eps_m);
	double tmp;
	if (x <= -0.41) {
		tmp = t_0 * (t_1 * -0.08333333333333333);
	} else if (x <= 3.1e-33) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 7.4e+169) {
		tmp = (t_1 * 0.08333333333333333) * t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (x * x)
	t_1 = eps_m * (eps_m * eps_m)
	tmp = 0
	if x <= -0.41:
		tmp = t_0 * (t_1 * -0.08333333333333333)
	elif x <= 3.1e-33:
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))))
	elif x <= 7.4e+169:
		tmp = (t_1 * 0.08333333333333333) * t_0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(eps_m * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= -0.41)
		tmp = Float64(t_0 * Float64(t_1 * -0.08333333333333333));
	elseif (x <= 3.1e-33)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.25 * Float64(eps_m * eps_m)))));
	elseif (x <= 7.4e+169)
		tmp = Float64(Float64(t_1 * 0.08333333333333333) * t_0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (x * x);
	t_1 = eps_m * (eps_m * eps_m);
	tmp = 0.0;
	if (x <= -0.41)
		tmp = t_0 * (t_1 * -0.08333333333333333);
	elseif (x <= 3.1e-33)
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	elseif (x <= 7.4e+169)
		tmp = (t_1 * 0.08333333333333333) * t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.41], N[(t$95$0 * N[(t$95$1 * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e-33], N[(1.0 + N[(x * N[(x * N[(0.25 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.4e+169], N[(N[(t$95$1 * 0.08333333333333333), $MachinePrecision] * t$95$0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.409999999999999976

   1. Initial program 94.7%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 48.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 48.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)}, \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\varepsilon}\right), \frac{-1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   10. Simplified 39.8%

       \[\leadsto \left(0.5 + \frac{0.5}{\varepsilon}\right) - \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot \left(0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(x \cdot -0.16666666666666666\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       13. *-lowering-*.f64 39.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -0.409999999999999976 < x < 3.09999999999999997e-33

   1. Initial program 54.5%

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

   3. Simplified 54.5%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 42.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 42.2%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 86.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 86.8%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 86.8%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 87.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   16. Simplified 87.3%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   if 3.09999999999999997e-33 < x < 7.40000000000000001e169

   1. Initial program 95.9%

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

   3. Simplified 95.9%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)}, \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)}\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - 1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \color{blue}{\mathsf{\_.f64}\left(-1, \varepsilon\right)}\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + -1\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + \left(\varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(-1 + \varepsilon\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + -1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\varepsilon - 1\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon - 1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \color{blue}{\varepsilon}\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\varepsilon + -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \left(x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\varepsilon, -1\right), \mathsf{*.f64}\left(x, \left(\frac{1}{6} \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{3}\right) + \frac{1}{2} \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{2}, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 29.4%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\left(\varepsilon + -1\right) + x \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right)\right)\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      13. *-lowering-*.f64 43.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 43.8%

       \[\leadsto \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if 7.40000000000000001e169 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.41:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-33}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+169}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 0.08333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;x \leq 215:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+171}:\\ \;\;\;\;\left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.8)
   (* (* x (* x x)) (* (* eps_m (* eps_m eps_m)) -0.08333333333333333))
   (if (<= x 215.0)
     (+ 1.0 (* x (* x (* 0.25 (* eps_m eps_m)))))
     (if (<= x 1.8e+171) (* (* 0.5 (* eps_m eps_m)) (* x x)) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.8) {
		tmp = (x * (x * x)) * ((eps_m * (eps_m * eps_m)) * -0.08333333333333333);
	} else if (x <= 215.0) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 1.8e+171) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.8d0)) then
        tmp = (x * (x * x)) * ((eps_m * (eps_m * eps_m)) * (-0.08333333333333333d0))
    else if (x <= 215.0d0) then
        tmp = 1.0d0 + (x * (x * (0.25d0 * (eps_m * eps_m))))
    else if (x <= 1.8d+171) then
        tmp = (0.5d0 * (eps_m * eps_m)) * (x * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.8) {
		tmp = (x * (x * x)) * ((eps_m * (eps_m * eps_m)) * -0.08333333333333333);
	} else if (x <= 215.0) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 1.8e+171) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.8:
		tmp = (x * (x * x)) * ((eps_m * (eps_m * eps_m)) * -0.08333333333333333)
	elif x <= 215.0:
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))))
	elif x <= 1.8e+171:
		tmp = (0.5 * (eps_m * eps_m)) * (x * x)
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.8)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(Float64(eps_m * Float64(eps_m * eps_m)) * -0.08333333333333333));
	elseif (x <= 215.0)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.25 * Float64(eps_m * eps_m)))));
	elseif (x <= 1.8e+171)
		tmp = Float64(Float64(0.5 * Float64(eps_m * eps_m)) * Float64(x * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.8)
		tmp = (x * (x * x)) * ((eps_m * (eps_m * eps_m)) * -0.08333333333333333);
	elseif (x <= 215.0)
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	elseif (x <= 1.8e+171)
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.8], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 215.0], N[(1.0 + N[(x * N[(x * N[(0.25 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+171], N[(N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.80000000000000004

   1. Initial program 94.7%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 48.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 48.9%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)}, \frac{-1}{2}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\varepsilon}\right), \frac{-1}{2}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. distribute-lft-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right) + \frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2} + \frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2}\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(1 + \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \varepsilon\right)\right)\right), \left(\frac{-1}{6} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{3}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   10. Simplified 39.8%

       \[\leadsto \left(0.5 + \frac{0.5}{\varepsilon}\right) - \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + x \cdot \left(0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(x \cdot -0.16666666666666666\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]

   11. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{\frac{-1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{3}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{3}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{3}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{3}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{3}\right)\right) \]

       9. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       13. *-lowering-*.f64 39.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 39.8%

       \[\leadsto \color{blue}{\left(-0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -0.80000000000000004 < x < 215

   1. Initial program 54.4%

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

   3. Simplified 54.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 41.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 41.7%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 85.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 85.0%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 85.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 85.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   16. Simplified 85.7%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   if 215 < x < 1.80000000000000009e171

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 3.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 64.4%

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

   if 1.80000000000000009e171 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.8:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.08333333333333333\right)\\ \mathbf{elif}\;x \leq 215:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+171}:\\ \;\;\;\;\left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 78.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 250:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+170}:\\ \;\;\;\;\left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* x (* x x)) (* (* eps_m eps_m) -0.3333333333333333))
   (if (<= x 250.0)
     (+ 1.0 (* x (* x (* 0.25 (* eps_m eps_m)))))
     (if (<= x 1.9e+170) (* (* 0.5 (* eps_m eps_m)) (* x x)) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333);
	} else if (x <= 250.0) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 1.9e+170) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * (x * x)) * ((eps_m * eps_m) * (-0.3333333333333333d0))
    else if (x <= 250.0d0) then
        tmp = 1.0d0 + (x * (x * (0.25d0 * (eps_m * eps_m))))
    else if (x <= 1.9d+170) then
        tmp = (0.5d0 * (eps_m * eps_m)) * (x * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333);
	} else if (x <= 250.0) {
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	} else if (x <= 1.9e+170) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333)
	elif x <= 250.0:
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))))
	elif x <= 1.9e+170:
		tmp = (0.5 * (eps_m * eps_m)) * (x * x)
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(Float64(eps_m * eps_m) * -0.3333333333333333));
	elseif (x <= 250.0)
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(0.25 * Float64(eps_m * eps_m)))));
	elseif (x <= 1.9e+170)
		tmp = Float64(Float64(0.5 * Float64(eps_m * eps_m)) * Float64(x * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333);
	elseif (x <= 250.0)
		tmp = 1.0 + (x * (x * (0.25 * (eps_m * eps_m))));
	elseif (x <= 1.9e+170)
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 250.0], N[(1.0 + N[(x * N[(x * N[(0.25 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e+170], N[(N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 94.7%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 14.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 80.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{2} \cdot \frac{-1}{3}\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{-1}{3}\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-1}{3}\right), \left({x}^{3}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \left({x}^{3}\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       11. *-lowering-*.f64 80.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 80.0%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.3333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -1 < x < 250

   1. Initial program 54.4%

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

   3. Simplified 54.4%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \frac{1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2} \cdot 1}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{2}}{\varepsilon}\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

      4. /-lowering-/.f64 41.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 41.7%

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

   8. Taylor expanded in eps around inf

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

      1. sub-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2} \cdot e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}\right)\right)}\right) \]

      11. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot \color{blue}{e^{x \cdot \left(-1 \cdot \varepsilon - 1\right)}}\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(-1 \cdot \varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\left(\mathsf{neg}\left(\varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(\varepsilon + 1\right)\right)\right)}\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)}\right)\right) \]

      16. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

      17. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right) \]

   10. Simplified 85.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right) + \frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \left(1 + \varepsilon\right)\right), \color{blue}{\left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(1 + \varepsilon\right) \cdot \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(1 + \varepsilon\right), \frac{-1}{2}\right), \left(\color{blue}{\frac{1}{4}} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\frac{1}{4} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \left(\left(\frac{1}{4} \cdot x\right) \cdot \color{blue}{{\left(1 + \varepsilon\right)}^{2}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot x\right), \color{blue}{\left({\left(1 + \varepsilon\right)}^{2}\right)}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left({\color{blue}{\left(1 + \varepsilon\right)}}^{2}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \color{blue}{\left(1 + \varepsilon\right)}\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \left(\color{blue}{1} + \varepsilon\right)\right)\right)\right)\right)\right) \]

       13. +-lowering-+.f64 85.0%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, x\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \varepsilon\right), \mathsf{+.f64}\left(1, \color{blue}{\varepsilon}\right)\right)\right)\right)\right)\right) \]

   13. Simplified 85.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot -0.5 + \left(0.25 \cdot x\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right)} \]

   14. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   15. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 85.7%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   16. Simplified 85.7%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]

   if 250 < x < 1.8999999999999999e170

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 3.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 64.4%

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

   if 1.8999999999999999e170 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 250:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+170}:\\ \;\;\;\;\left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 72.5% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;1 + x \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+169}:\\ \;\;\;\;\left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.38e-11)
   (* (* x (* x x)) (* (* eps_m eps_m) -0.3333333333333333))
   (if (<= x 0.0034)
     (+ 1.0 (* x (* x -0.5)))
     (if (<= x 1.3e+169) (* (* 0.5 (* eps_m eps_m)) (* x x)) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.38e-11) {
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333);
	} else if (x <= 0.0034) {
		tmp = 1.0 + (x * (x * -0.5));
	} else if (x <= 1.3e+169) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.38d-11)) then
        tmp = (x * (x * x)) * ((eps_m * eps_m) * (-0.3333333333333333d0))
    else if (x <= 0.0034d0) then
        tmp = 1.0d0 + (x * (x * (-0.5d0)))
    else if (x <= 1.3d+169) then
        tmp = (0.5d0 * (eps_m * eps_m)) * (x * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.38e-11) {
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333);
	} else if (x <= 0.0034) {
		tmp = 1.0 + (x * (x * -0.5));
	} else if (x <= 1.3e+169) {
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.38e-11:
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333)
	elif x <= 0.0034:
		tmp = 1.0 + (x * (x * -0.5))
	elif x <= 1.3e+169:
		tmp = (0.5 * (eps_m * eps_m)) * (x * x)
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.38e-11)
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(Float64(eps_m * eps_m) * -0.3333333333333333));
	elseif (x <= 0.0034)
		tmp = Float64(1.0 + Float64(x * Float64(x * -0.5)));
	elseif (x <= 1.3e+169)
		tmp = Float64(Float64(0.5 * Float64(eps_m * eps_m)) * Float64(x * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.38e-11)
		tmp = (x * (x * x)) * ((eps_m * eps_m) * -0.3333333333333333);
	elseif (x <= 0.0034)
		tmp = 1.0 + (x * (x * -0.5));
	elseif (x <= 1.3e+169)
		tmp = (0.5 * (eps_m * eps_m)) * (x * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.38e-11], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0034], N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.3e+169], N[(N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.38e-11

   1. Initial program 92.8%

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

   3. Simplified 92.8%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 14.4%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 76.8%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around inf

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{2} \cdot \frac{-1}{3}\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \frac{-1}{3}\right), \left({\color{blue}{x}}^{3}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{-1}{3}\right), \left({x}^{3}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \left({x}^{3}\right)\right) \]

       7. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       11. *-lowering-*.f64 76.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{-1}{3}\right), \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   13. Simplified 76.9%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.3333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -1.38e-11 < x < 0.00339999999999999981

   1. Initial program 54.2%

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

   3. Simplified 54.2%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 63.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\varepsilon \cdot \left(x \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + x \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + x \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   10. Simplified 74.0%

       \[\leadsto 1 + x \cdot \color{blue}{\frac{x \cdot \left(0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + \left(x \cdot 0.08333333333333333 - 0.25\right)\right) + \varepsilon \cdot \left(x \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{-1}{2} \cdot x\right) \cdot x\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       7. *-lowering-*.f64 73.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   13. Simplified 73.8%

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

   if 0.00339999999999999981 < x < 1.3e169

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 3.7%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 0.3%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f64 64.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 64.4%

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

   if 1.3e169 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38 \cdot 10^{-11}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;x \leq 0.0034:\\ \;\;\;\;1 + x \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+169}:\\ \;\;\;\;\left(0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 72.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(0.5 \cdot \left(eps\_m \cdot eps\_m\right)\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.012:\\ \;\;\;\;1 + x \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+170}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (* eps_m eps_m)) (* x x))))
   (if (<= x -3.4e-10)
     t_0
     (if (<= x 0.012) (+ 1.0 (* x (* x -0.5))) (if (<= x 8.2e+170) t_0 0.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (0.5 * (eps_m * eps_m)) * (x * x);
	double tmp;
	if (x <= -3.4e-10) {
		tmp = t_0;
	} else if (x <= 0.012) {
		tmp = 1.0 + (x * (x * -0.5));
	} else if (x <= 8.2e+170) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 * (eps_m * eps_m)) * (x * x)
    if (x <= (-3.4d-10)) then
        tmp = t_0
    else if (x <= 0.012d0) then
        tmp = 1.0d0 + (x * (x * (-0.5d0)))
    else if (x <= 8.2d+170) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (0.5 * (eps_m * eps_m)) * (x * x);
	double tmp;
	if (x <= -3.4e-10) {
		tmp = t_0;
	} else if (x <= 0.012) {
		tmp = 1.0 + (x * (x * -0.5));
	} else if (x <= 8.2e+170) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (0.5 * (eps_m * eps_m)) * (x * x)
	tmp = 0
	if x <= -3.4e-10:
		tmp = t_0
	elif x <= 0.012:
		tmp = 1.0 + (x * (x * -0.5))
	elif x <= 8.2e+170:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(0.5 * Float64(eps_m * eps_m)) * Float64(x * x))
	tmp = 0.0
	if (x <= -3.4e-10)
		tmp = t_0;
	elseif (x <= 0.012)
		tmp = Float64(1.0 + Float64(x * Float64(x * -0.5)));
	elseif (x <= 8.2e+170)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (0.5 * (eps_m * eps_m)) * (x * x);
	tmp = 0.0;
	if (x <= -3.4e-10)
		tmp = t_0;
	elseif (x <= 0.012)
		tmp = 1.0 + (x * (x * -0.5));
	elseif (x <= 8.2e+170)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(0.5 * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-10], t$95$0, If[LessEqual[x, 0.012], N[(1.0 + N[(x * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e+170], t$95$0, 0.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.40000000000000015e-10 or 0.012 < x < 8.2000000000000001e170

   1. Initial program 96.5%

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

   3. Simplified 96.5%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 8.9%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)} + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}{\varepsilon}\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \left(\frac{-1}{12} + \frac{1}{12}\right)\right) + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right) + \frac{{x}^{2} \cdot \left(\frac{1}{4} + \color{blue}{\left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}\right)}{\varepsilon}\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + \frac{{x}^{2} \cdot \color{blue}{\left(\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)}}{\varepsilon}\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0 + {x}^{2} \cdot \color{blue}{\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}}\right)\right) \]

      11. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(0 + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{0} + \frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(0, \color{blue}{\left(\frac{\frac{1}{4} + \left(\frac{-1}{6} \cdot x + \left(\frac{-1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)}{\varepsilon}\right)}\right)\right)\right) \]

   10. Simplified 37.7%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0 + \frac{0.25 + \left(x \cdot -0.25 + 0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot {\varepsilon}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\varepsilon}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f64 65.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   13. Simplified 65.0%

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

   if -3.40000000000000015e-10 < x < 0.012

   1. Initial program 54.2%

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

   3. Simplified 54.2%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 63.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\varepsilon \cdot \left(x \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + x \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + x \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   10. Simplified 74.0%

       \[\leadsto 1 + x \cdot \color{blue}{\frac{x \cdot \left(0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + \left(x \cdot 0.08333333333333333 - 0.25\right)\right) + \varepsilon \cdot \left(x \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in x around 0

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(\frac{-1}{2} \cdot x\right) \cdot x\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

       7. *-lowering-*.f64 73.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   13. Simplified 73.8%

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

   if 8.2000000000000001e170 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.2%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 0.0%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.1%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.1%

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

Alternative 20: 57.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 480000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+94}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+170}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 480000000.0)
   1.0
   (if (<= x 7.5e+94)
     0.0
     (if (<= x 8e+170) (* (* x (* x x)) 0.3333333333333333) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 480000000.0) {
		tmp = 1.0;
	} else if (x <= 7.5e+94) {
		tmp = 0.0;
	} else if (x <= 8e+170) {
		tmp = (x * (x * x)) * 0.3333333333333333;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 480000000.0d0) then
        tmp = 1.0d0
    else if (x <= 7.5d+94) then
        tmp = 0.0d0
    else if (x <= 8d+170) then
        tmp = (x * (x * x)) * 0.3333333333333333d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 480000000.0) {
		tmp = 1.0;
	} else if (x <= 7.5e+94) {
		tmp = 0.0;
	} else if (x <= 8e+170) {
		tmp = (x * (x * x)) * 0.3333333333333333;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 480000000.0:
		tmp = 1.0
	elif x <= 7.5e+94:
		tmp = 0.0
	elif x <= 8e+170:
		tmp = (x * (x * x)) * 0.3333333333333333
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 480000000.0)
		tmp = 1.0;
	elseif (x <= 7.5e+94)
		tmp = 0.0;
	elseif (x <= 8e+170)
		tmp = Float64(Float64(x * Float64(x * x)) * 0.3333333333333333);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 480000000.0)
		tmp = 1.0;
	elseif (x <= 7.5e+94)
		tmp = 0.0;
	elseif (x <= 8e+170)
		tmp = (x * (x * x)) * 0.3333333333333333;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 480000000.0], 1.0, If[LessEqual[x, 7.5e+94], 0.0, If[LessEqual[x, 8e+170], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.8e8

   1. Initial program 63.2%

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

   3. Simplified 63.2%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 57.3%

      \[\leadsto \color{blue}{1} \]

   if 4.8e8 < x < 7.49999999999999978e94 or 8.00000000000000028e170 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 0.6%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 23.9%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 62.5%

          \[\leadsto 0 \]

   12. Applied egg-rr 62.5%

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

   if 7.49999999999999978e94 < x < 8.00000000000000028e170

   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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 5.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around 0

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{\varepsilon \cdot \left(x \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + x \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)}{\varepsilon}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\varepsilon \cdot \left(x \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\frac{1}{12} \cdot x + \frac{1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right) + x \cdot \left(\left(\frac{1}{12} \cdot x + \frac{1}{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) - \frac{1}{4}\right)\right), \color{blue}{\varepsilon}\right)\right)\right) \]

   10. Simplified 67.9%

       \[\leadsto 1 + x \cdot \color{blue}{\frac{x \cdot \left(0.5 \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + \left(x \cdot 0.08333333333333333 - 0.25\right)\right) + \varepsilon \cdot \left(x \cdot \left(\left(-0.25 + -0.5 \cdot \left(x \cdot -0.16666666666666666\right)\right) + \left(x \cdot 0.25 + -0.25\right)\right)\right)}{\varepsilon}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{3} \cdot {x}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{\frac{1}{3}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\frac{1}{3}}\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(x \cdot x\right)\right), \frac{1}{3}\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{2}\right), \frac{1}{3}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \frac{1}{3}\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot x\right)\right), \frac{1}{3}\right) \]

       7. *-lowering-*.f64 67.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right), \frac{1}{3}\right) \]

   13. Simplified 67.5%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.3333333333333333} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 57.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 480000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 480000000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 480000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 480000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 480000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 480000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 480000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 480000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 480000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.8e8

   1. Initial program 63.2%

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

   3. Simplified 63.2%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 57.3%

      \[\leadsto \color{blue}{1} \]

   if 4.8e8 < 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%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 2.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

      9. mul0-rgt N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

      12. *-lowering-*.f64 24.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

   10. Simplified 24.4%

       \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

       2. mul0-rgt 50.8%

          \[\leadsto 0 \]

   12. Applied egg-rr 50.8%

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

Alternative 22: 16.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

   \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

7. Simplified 38.9%

   \[\leadsto \color{blue}{1 + x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) + \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(\left(x \cdot 0.16666666666666666\right) \cdot \left(\varepsilon + -1\right) + 0.5\right)\right) + x \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(0.5 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(-1 - \varepsilon\right)\right)\right)\right)\right)} \]

8. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{{\varepsilon}^{3} \cdot \left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)}\right) \]

   2. cube-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{{x}^{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{\color{blue}{2}} \cdot \left(\frac{-1}{12} \cdot x + \frac{1}{12} \cdot x\right)\right)\right) \]

   7. distribute-rgt-out N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot \color{blue}{\left(\frac{-1}{12} + \frac{1}{12}\right)}\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot \left(x \cdot 0\right)\right)\right) \]

   9. mul0-rgt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left({x}^{2} \cdot 0\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{0}\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), 0\right)\right) \]

   12. *-lowering-*.f64 7.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), 0\right)\right) \]

10. Simplified 7.5%

    \[\leadsto \color{blue}{\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right)} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0} \]

    2. mul0-rgt 14.9%

       \[\leadsto 0 \]

12. Applied egg-rr 14.9%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))