NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 99.8%

Time: 16.7s

Alternatives: 15

Speedup: 11.9×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00037:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - eps\_m\right)} + e^{x \cdot eps\_m}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00037)
   (* (+ x 1.0) (exp (- 0.0 x)))
   (* 0.5 (+ (exp (* x (- -1.0 eps_m))) (exp (* x eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00037) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((x * 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 <= 0.00037d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else
        tmp = 0.5d0 * (exp((x * ((-1.0d0) - eps_m))) + exp((x * 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 <= 0.00037) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else {
		tmp = 0.5 * (Math.exp((x * (-1.0 - eps_m))) + Math.exp((x * eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00037:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	else:
		tmp = 0.5 * (math.exp((x * (-1.0 - eps_m))) + math.exp((x * eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00037)
		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(x * eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00037)
		tmp = (x + 1.0) * exp((0.0 - x));
	else
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((x * eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00037], 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[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.6999999999999999e-4

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in 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 78.9%

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

   7. Simplified 78.9%

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

   if 3.6999999999999999e-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 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00037:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0.5 \cdot \left(e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-1 - eps\_m\right)}\right) \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.5 * (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - eps_m))));
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.5d0 * (exp((x * (eps_m + (-1.0d0)))) + exp((x * ((-1.0d0) - eps_m))))
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.5 * (math.exp((x * (eps_m + -1.0))) + math.exp((x * (-1.0 - eps_m))))

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(0.5 * Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-1.0 - eps_m)))))
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.5 * (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - eps_m))));
end

Wolfram

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

Derivation

1. Initial program 69.0%

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

3. Simplified 69.0%

   \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
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 98.9%

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

7. Simplified 98.9%

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

Alternative 3: 95.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{eps\_m}\\ \mathbf{if}\;eps\_m \leq 0.00037:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(\frac{0.5}{eps\_m} + x \cdot \left(\left(eps\_m + -1\right) \cdot t\_0 + 0.5 \cdot \left(\left(eps\_m + -1\right) \cdot \left(\left(x \cdot \left(eps\_m + -1\right)\right) \cdot t\_0\right)\right)\right)\right)\right) - e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 eps_m))))
   (if (<= eps_m 0.00037)
     (* (+ x 1.0) (exp (- 0.0 x)))
     (-
      (+
       0.5
       (+
        (/ 0.5 eps_m)
        (*
         x
         (+
          (* (+ eps_m -1.0) t_0)
          (* 0.5 (* (+ eps_m -1.0) (* (* x (+ eps_m -1.0)) t_0)))))))
      (* (exp (* x (- -1.0 eps_m))) (+ (/ 0.5 eps_m) -0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.5 + (0.5 / eps_m);
	double tmp;
	if (eps_m <= 0.00037) {
		tmp = (x + 1.0) * exp((0.0 - x));
	} else {
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) - (exp((x * (-1.0 - eps_m))) * ((0.5 / eps_m) + -0.5));
	}
	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 + (0.5d0 / eps_m)
    if (eps_m <= 0.00037d0) then
        tmp = (x + 1.0d0) * exp((0.0d0 - x))
    else
        tmp = (0.5d0 + ((0.5d0 / eps_m) + (x * (((eps_m + (-1.0d0)) * t_0) + (0.5d0 * ((eps_m + (-1.0d0)) * ((x * (eps_m + (-1.0d0))) * t_0))))))) - (exp((x * ((-1.0d0) - eps_m))) * ((0.5d0 / eps_m) + (-0.5d0)))
    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 + (0.5 / eps_m);
	double tmp;
	if (eps_m <= 0.00037) {
		tmp = (x + 1.0) * Math.exp((0.0 - x));
	} else {
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) - (Math.exp((x * (-1.0 - eps_m))) * ((0.5 / eps_m) + -0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 0.5 + (0.5 / eps_m)
	tmp = 0
	if eps_m <= 0.00037:
		tmp = (x + 1.0) * math.exp((0.0 - x))
	else:
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) - (math.exp((x * (-1.0 - eps_m))) * ((0.5 / eps_m) + -0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.5 + Float64(0.5 / eps_m))
	tmp = 0.0
	if (eps_m <= 0.00037)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(0.0 - x)));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.5 / eps_m) + Float64(x * Float64(Float64(Float64(eps_m + -1.0) * t_0) + Float64(0.5 * Float64(Float64(eps_m + -1.0) * Float64(Float64(x * Float64(eps_m + -1.0)) * t_0))))))) - Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(Float64(0.5 / eps_m) + -0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 0.5 + (0.5 / eps_m);
	tmp = 0.0;
	if (eps_m <= 0.00037)
		tmp = (x + 1.0) * exp((0.0 - x));
	else
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) - (exp((x * (-1.0 - eps_m))) * ((0.5 / eps_m) + -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.00037], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(N[(0.5 / eps$95$m), $MachinePrecision] + N[(x * N[(N[(N[(eps$95$m + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * N[(N[(eps$95$m + -1.0), $MachinePrecision] * N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.6999999999999999e-4

   1. Initial program 58.2%

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

   3. Simplified 58.2%

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

   5. Taylor expanded in 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 78.9%

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

   7. Simplified 78.9%

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

   if 3.6999999999999999e-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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 91.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00037:\\ \;\;\;\;\left(x + 1\right) \cdot e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(\frac{0.5}{\varepsilon} + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(x \cdot \left(\varepsilon + -1\right)\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right)\right)\right)\right)\right)\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.5 + \frac{0.5}{eps\_m}\\ \mathbf{if}\;eps\_m \leq 7.8 \cdot 10^{+57}:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(\frac{0.5}{eps\_m} + x \cdot \left(\left(eps\_m + -1\right) \cdot t\_0 + 0.5 \cdot \left(\left(eps\_m + -1\right) \cdot \left(\left(x \cdot \left(eps\_m + -1\right)\right) \cdot t\_0\right)\right)\right)\right)\right) + \left(\frac{0.5}{eps\_m} + -0.5\right) \cdot \left(-1 + x \cdot \left(\left(eps\_m + 1\right) + \left(-1 - eps\_m\right) \cdot \left(\left(0.5 \cdot x\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 0.5 (/ 0.5 eps_m))))
   (if (<= eps_m 7.8e+57)
     (exp (- 0.0 x))
     (+
      (+
       0.5
       (+
        (/ 0.5 eps_m)
        (*
         x
         (+
          (* (+ eps_m -1.0) t_0)
          (* 0.5 (* (+ eps_m -1.0) (* (* x (+ eps_m -1.0)) t_0)))))))
      (*
       (+ (/ 0.5 eps_m) -0.5)
       (+
        -1.0
        (*
         x
         (+
          (+ eps_m 1.0)
          (* (- -1.0 eps_m) (* (* 0.5 x) (+ eps_m 1.0)))))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.5 + (0.5 / eps_m);
	double tmp;
	if (eps_m <= 7.8e+57) {
		tmp = exp((0.0 - x));
	} else {
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) + (((0.5 / eps_m) + -0.5) * (-1.0 + (x * ((eps_m + 1.0) + ((-1.0 - eps_m) * ((0.5 * x) * (eps_m + 1.0)))))));
	}
	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 + (0.5d0 / eps_m)
    if (eps_m <= 7.8d+57) then
        tmp = exp((0.0d0 - x))
    else
        tmp = (0.5d0 + ((0.5d0 / eps_m) + (x * (((eps_m + (-1.0d0)) * t_0) + (0.5d0 * ((eps_m + (-1.0d0)) * ((x * (eps_m + (-1.0d0))) * t_0))))))) + (((0.5d0 / eps_m) + (-0.5d0)) * ((-1.0d0) + (x * ((eps_m + 1.0d0) + (((-1.0d0) - eps_m) * ((0.5d0 * x) * (eps_m + 1.0d0)))))))
    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 + (0.5 / eps_m);
	double tmp;
	if (eps_m <= 7.8e+57) {
		tmp = Math.exp((0.0 - x));
	} else {
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) + (((0.5 / eps_m) + -0.5) * (-1.0 + (x * ((eps_m + 1.0) + ((-1.0 - eps_m) * ((0.5 * x) * (eps_m + 1.0)))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 0.5 + (0.5 / eps_m)
	tmp = 0
	if eps_m <= 7.8e+57:
		tmp = math.exp((0.0 - x))
	else:
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) + (((0.5 / eps_m) + -0.5) * (-1.0 + (x * ((eps_m + 1.0) + ((-1.0 - eps_m) * ((0.5 * x) * (eps_m + 1.0)))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.5 + Float64(0.5 / eps_m))
	tmp = 0.0
	if (eps_m <= 7.8e+57)
		tmp = exp(Float64(0.0 - x));
	else
		tmp = Float64(Float64(0.5 + Float64(Float64(0.5 / eps_m) + Float64(x * Float64(Float64(Float64(eps_m + -1.0) * t_0) + Float64(0.5 * Float64(Float64(eps_m + -1.0) * Float64(Float64(x * Float64(eps_m + -1.0)) * t_0))))))) + Float64(Float64(Float64(0.5 / eps_m) + -0.5) * Float64(-1.0 + Float64(x * Float64(Float64(eps_m + 1.0) + Float64(Float64(-1.0 - eps_m) * Float64(Float64(0.5 * x) * Float64(eps_m + 1.0))))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 0.5 + (0.5 / eps_m);
	tmp = 0.0;
	if (eps_m <= 7.8e+57)
		tmp = exp((0.0 - x));
	else
		tmp = (0.5 + ((0.5 / eps_m) + (x * (((eps_m + -1.0) * t_0) + (0.5 * ((eps_m + -1.0) * ((x * (eps_m + -1.0)) * t_0))))))) + (((0.5 / eps_m) + -0.5) * (-1.0 + (x * ((eps_m + 1.0) + ((-1.0 - eps_m) * ((0.5 * x) * (eps_m + 1.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 7.8e+57], N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision], N[(N[(0.5 + N[(N[(0.5 / eps$95$m), $MachinePrecision] + N[(x * N[(N[(N[(eps$95$m + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 * N[(N[(eps$95$m + -1.0), $MachinePrecision] * N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * N[(-1.0 + N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] + N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(N[(0.5 * x), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.79999999999999937e57

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in 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 98.6%

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

   7. Simplified 98.6%

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

   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 87.1%

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

   10. Simplified 87.1%

       \[\leadsto \color{blue}{e^{0 - x}} \]
   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 87.1%

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

   12. Applied egg-rr 87.1%

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

   if 7.79999999999999937e57 < 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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 89.5%

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

   8. Taylor expanded in x around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

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

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

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

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

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

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. unpow2 N/A

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

      15. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      20. +-lowering-+.f64 84.4%

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

   10. Simplified 84.4%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7.8 \cdot 10^{+57}:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \left(\frac{0.5}{\varepsilon} + x \cdot \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(x \cdot \left(\varepsilon + -1\right)\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right)\right)\right)\right)\right)\right) + \left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(-1 + x \cdot \left(\left(\varepsilon + 1\right) + \left(-1 - \varepsilon\right) \cdot \left(\left(0.5 \cdot x\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-132}:\\ \;\;\;\;1 + x \cdot \left(\left(-0.08333333333333333 \cdot \left(eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;1 + eps\_m \cdot \left(x \cdot \left(eps\_m \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{eps\_m}\right) - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.8e-132)
   (+ 1.0 (* x (* (* -0.08333333333333333 (* eps_m (* eps_m eps_m))) (* x x))))
   (if (<= x 1.5)
     (+ 1.0 (* eps_m (* x (* eps_m (* x (+ 0.5 (* x -0.3333333333333333)))))))
     (if (<= x 3.4e+115)
       (* x (* 0.25 (* x (* eps_m eps_m))))
       (- (+ 0.5 (/ 0.5 eps_m)) (+ (/ 0.5 eps_m) -0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.8e-132) {
		tmp = 1.0 + (x * ((-0.08333333333333333 * (eps_m * (eps_m * eps_m))) * (x * x)));
	} else if (x <= 1.5) {
		tmp = 1.0 + (eps_m * (x * (eps_m * (x * (0.5 + (x * -0.3333333333333333))))));
	} else if (x <= 3.4e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	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 <= (-1.8d-132)) then
        tmp = 1.0d0 + (x * (((-0.08333333333333333d0) * (eps_m * (eps_m * eps_m))) * (x * x)))
    else if (x <= 1.5d0) then
        tmp = 1.0d0 + (eps_m * (x * (eps_m * (x * (0.5d0 + (x * (-0.3333333333333333d0)))))))
    else if (x <= 3.4d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 + (0.5d0 / eps_m)) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= -1.8e-132) {
		tmp = 1.0 + (x * ((-0.08333333333333333 * (eps_m * (eps_m * eps_m))) * (x * x)));
	} else if (x <= 1.5) {
		tmp = 1.0 + (eps_m * (x * (eps_m * (x * (0.5 + (x * -0.3333333333333333))))));
	} else if (x <= 3.4e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.8e-132:
		tmp = 1.0 + (x * ((-0.08333333333333333 * (eps_m * (eps_m * eps_m))) * (x * x)))
	elif x <= 1.5:
		tmp = 1.0 + (eps_m * (x * (eps_m * (x * (0.5 + (x * -0.3333333333333333))))))
	elif x <= 3.4e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.8e-132)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(-0.08333333333333333 * Float64(eps_m * Float64(eps_m * eps_m))) * Float64(x * x))));
	elseif (x <= 1.5)
		tmp = Float64(1.0 + Float64(eps_m * Float64(x * Float64(eps_m * Float64(x * Float64(0.5 + Float64(x * -0.3333333333333333)))))));
	elseif (x <= 3.4e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.5 / eps_m)) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.8e-132)
		tmp = 1.0 + (x * ((-0.08333333333333333 * (eps_m * (eps_m * eps_m))) * (x * x)));
	elseif (x <= 1.5)
		tmp = 1.0 + (eps_m * (x * (eps_m * (x * (0.5 + (x * -0.3333333333333333))))));
	elseif (x <= 3.4e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.8e-132], N[(1.0 + N[(x * N[(N[(-0.08333333333333333 * N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5], N[(1.0 + N[(eps$95$m * N[(x * N[(eps$95$m * N[(x * N[(0.5 + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.80000000000000004e-132

   1. Initial program 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 75.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{6} \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   10. Simplified 47.2%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\left(0.16666666666666666 \cdot x\right) \cdot \left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + 0.5 \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right)\right)\right) + \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{12} \cdot \left({\varepsilon}^{3} \cdot {x}^{2}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right) \cdot \color{blue}{{x}^{2}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\frac{-1}{12} \cdot {\varepsilon}^{3}\right), \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left({\varepsilon}^{3}\right)\right), \left({\color{blue}{x}}^{2}\right)\right)\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{2}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \left(\varepsilon \cdot {\varepsilon}^{2}\right)\right), \left({x}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right)\right), \left({x}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right)\right), \left({x}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left({x}^{2}\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 57.4%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   13. Simplified 57.4%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(-0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot x\right)\right)} \]

   if -1.80000000000000004e-132 < x < 1.5

   1. Initial program 44.0%

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

   3. Simplified 44.0%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   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 68.5%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right) + \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.6%

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

   10. Simplified 88.6%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

       9. *-lowering-*.f64 89.2%

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

   12. Applied egg-rr 89.2%

       \[\leadsto 1 + \color{blue}{\varepsilon \cdot \left(\left(\varepsilon \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right) \cdot x\right)} \]

   if 1.5 < x < 3.4000000000000001e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 3.4000000000000001e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-132}:\\ \;\;\;\;1 + x \cdot \left(\left(-0.08333333333333333 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;1 + \varepsilon \cdot \left(x \cdot \left(\varepsilon \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{\varepsilon}\right) - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(eps\_m \cdot eps\_m\right) \cdot -0.3333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 0.06:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{eps\_m} - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.95e-44)
   (* (* (* eps_m eps_m) -0.3333333333333333) (* x (* x x)))
   (if (<= x 0.06)
     1.0
     (if (<= x 3.4e+115)
       (* x (* 0.25 (* x (* eps_m eps_m))))
       (- (/ 0.5 eps_m) (+ (/ 0.5 eps_m) -0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.95e-44) {
		tmp = ((eps_m * eps_m) * -0.3333333333333333) * (x * (x * x));
	} else if (x <= 0.06) {
		tmp = 1.0;
	} else if (x <= 3.4e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	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 <= (-1.95d-44)) then
        tmp = ((eps_m * eps_m) * (-0.3333333333333333d0)) * (x * (x * x))
    else if (x <= 0.06d0) then
        tmp = 1.0d0
    else if (x <= 3.4d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 / eps_m) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= -1.95e-44) {
		tmp = ((eps_m * eps_m) * -0.3333333333333333) * (x * (x * x));
	} else if (x <= 0.06) {
		tmp = 1.0;
	} else if (x <= 3.4e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.95e-44:
		tmp = ((eps_m * eps_m) * -0.3333333333333333) * (x * (x * x))
	elif x <= 0.06:
		tmp = 1.0
	elif x <= 3.4e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.95e-44)
		tmp = Float64(Float64(Float64(eps_m * eps_m) * -0.3333333333333333) * Float64(x * Float64(x * x)));
	elseif (x <= 0.06)
		tmp = 1.0;
	elseif (x <= 3.4e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 / eps_m) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.95e-44)
		tmp = ((eps_m * eps_m) * -0.3333333333333333) * (x * (x * x));
	elseif (x <= 0.06)
		tmp = 1.0;
	elseif (x <= 3.4e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.95e-44], N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.06], 1.0, If[LessEqual[x, 3.4e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / eps$95$m), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9500000000000001e-44

   1. Initial program 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   7. Simplified 31.3%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right) + \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 90.2%

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

   10. Simplified 90.2%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)} \]

   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 81.3%

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

   13. Simplified 81.3%

       \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.3333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

   if -1.9500000000000001e-44 < x < 0.059999999999999998

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 83.4%

      \[\leadsto \color{blue}{1} \]

   if 0.059999999999999998 < x < 3.4000000000000001e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 3.4000000000000001e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 67.7%

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

   13. Simplified 67.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-44}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.3333333333333333\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 0.06:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\varepsilon} - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.021:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{eps\_m} - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* 0.25 (* x (* eps_m eps_m))))))
   (if (<= x -4.4e-44)
     t_0
     (if (<= x 0.021)
       1.0
       (if (<= x 2.6e+115) t_0 (- (/ 0.5 eps_m) (+ (/ 0.5 eps_m) -0.5)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (0.25 * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -4.4e-44) {
		tmp = t_0;
	} else if (x <= 0.021) {
		tmp = 1.0;
	} else if (x <= 2.6e+115) {
		tmp = t_0;
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	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 * (0.25d0 * (x * (eps_m * eps_m)))
    if (x <= (-4.4d-44)) then
        tmp = t_0
    else if (x <= 0.021d0) then
        tmp = 1.0d0
    else if (x <= 2.6d+115) then
        tmp = t_0
    else
        tmp = (0.5d0 / eps_m) - ((0.5d0 / eps_m) + (-0.5d0))
    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 * (0.25 * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -4.4e-44) {
		tmp = t_0;
	} else if (x <= 0.021) {
		tmp = 1.0;
	} else if (x <= 2.6e+115) {
		tmp = t_0;
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (0.25 * (x * (eps_m * eps_m)))
	tmp = 0
	if x <= -4.4e-44:
		tmp = t_0
	elif x <= 0.021:
		tmp = 1.0
	elif x <= 2.6e+115:
		tmp = t_0
	else:
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))))
	tmp = 0.0
	if (x <= -4.4e-44)
		tmp = t_0;
	elseif (x <= 0.021)
		tmp = 1.0;
	elseif (x <= 2.6e+115)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.5 / eps_m) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (0.25 * (x * (eps_m * eps_m)));
	tmp = 0.0;
	if (x <= -4.4e-44)
		tmp = t_0;
	elseif (x <= 0.021)
		tmp = 1.0;
	elseif (x <= 2.6e+115)
		tmp = t_0;
	else
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.4e-44], t$95$0, If[LessEqual[x, 0.021], 1.0, If[LessEqual[x, 2.6e+115], t$95$0, N[(N[(0.5 / eps$95$m), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000024e-44 or 0.0210000000000000013 < x < 2.6e115

   1. Initial program 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 71.0%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 73.8%

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

   10. Simplified 73.8%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if -4.40000000000000024e-44 < x < 0.0210000000000000013

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 83.4%

      \[\leadsto \color{blue}{1} \]

   if 2.6e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 67.7%

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

   13. Simplified 67.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.021:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\varepsilon} - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;1 + x \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.66 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{eps\_m}\right) - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 3.0)
   (+ 1.0 (* x (* eps_m (* eps_m (* x (+ 0.5 (* x -0.16666666666666666)))))))
   (if (<= x 1.66e+115)
     (* x (* 0.25 (* x (* eps_m eps_m))))
     (- (+ 0.5 (/ 0.5 eps_m)) (+ (/ 0.5 eps_m) -0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3.0) {
		tmp = 1.0 + (x * (eps_m * (eps_m * (x * (0.5 + (x * -0.16666666666666666))))));
	} else if (x <= 1.66e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	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 <= 3.0d0) then
        tmp = 1.0d0 + (x * (eps_m * (eps_m * (x * (0.5d0 + (x * (-0.16666666666666666d0)))))))
    else if (x <= 1.66d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 + (0.5d0 / eps_m)) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= 3.0) {
		tmp = 1.0 + (x * (eps_m * (eps_m * (x * (0.5 + (x * -0.16666666666666666))))));
	} else if (x <= 1.66e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3.0:
		tmp = 1.0 + (x * (eps_m * (eps_m * (x * (0.5 + (x * -0.16666666666666666))))))
	elif x <= 1.66e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(1.0 + Float64(x * Float64(eps_m * Float64(eps_m * Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))));
	elseif (x <= 1.66e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.5 / eps_m)) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3.0)
		tmp = 1.0 + (x * (eps_m * (eps_m * (x * (0.5 + (x * -0.16666666666666666))))));
	elseif (x <= 1.66e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3.0], N[(1.0 + N[(x * N[(eps$95$m * N[(eps$95$m * N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.66e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 51.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{6} \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   10. Simplified 55.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\left(0.16666666666666666 \cdot x\right) \cdot \left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + 0.5 \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right)\right)\right) + \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({\varepsilon}^{3} \cdot \left(\frac{-1}{12} \cdot {x}^{2} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)}\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(\frac{-1}{12} \cdot {x}^{2}\right), \color{blue}{\left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left({x}^{2} \cdot \frac{-1}{12}\right), \left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}}{\varepsilon}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{12}\right), \left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}}{\varepsilon}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{12}\right), \left(\frac{\color{blue}{x} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \left(\frac{\color{blue}{x} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right), \color{blue}{\varepsilon}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot x\right)\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{6}\right)\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 65.5%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

   13. Simplified 65.5%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333 + \frac{x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)}{\varepsilon}\right)\right)} \]

   14. Taylor expanded in eps around 0

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

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right)\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right) \cdot \varepsilon\right), \color{blue}{\varepsilon}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right), \varepsilon\right), \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right), \varepsilon\right), \varepsilon\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot x\right)\right)\right), \varepsilon\right), \varepsilon\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{6}\right)\right)\right), \varepsilon\right), \varepsilon\right)\right)\right) \]

       9. *-lowering-*.f64 89.9%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right), \varepsilon\right), \varepsilon\right)\right)\right) \]

   16. Simplified 89.9%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right)} \]

   if 3 < x < 1.66e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 1.66e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;1 + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.66 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{\varepsilon}\right) - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;1 + x \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{eps\_m}\right) - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.5)
   (+ 1.0 (* x (* (* eps_m eps_m) (* x (+ 0.5 (* x -0.3333333333333333))))))
   (if (<= x 2.1e+115)
     (* x (* 0.25 (* x (* eps_m eps_m))))
     (- (+ 0.5 (/ 0.5 eps_m)) (+ (/ 0.5 eps_m) -0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.5) {
		tmp = 1.0 + (x * ((eps_m * eps_m) * (x * (0.5 + (x * -0.3333333333333333)))));
	} else if (x <= 2.1e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	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 <= 1.5d0) then
        tmp = 1.0d0 + (x * ((eps_m * eps_m) * (x * (0.5d0 + (x * (-0.3333333333333333d0))))))
    else if (x <= 2.1d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 + (0.5d0 / eps_m)) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= 1.5) {
		tmp = 1.0 + (x * ((eps_m * eps_m) * (x * (0.5 + (x * -0.3333333333333333)))));
	} else if (x <= 2.1e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.5:
		tmp = 1.0 + (x * ((eps_m * eps_m) * (x * (0.5 + (x * -0.3333333333333333)))))
	elif x <= 2.1e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(eps_m * eps_m) * Float64(x * Float64(0.5 + Float64(x * -0.3333333333333333))))));
	elseif (x <= 2.1e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.5 / eps_m)) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = 1.0 + (x * ((eps_m * eps_m) * (x * (0.5 + (x * -0.3333333333333333)))));
	elseif (x <= 2.1e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.5], N[(1.0 + N[(x * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * N[(0.5 + N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.5

   1. Initial program 56.8%

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

   3. Simplified 56.8%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   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 57.9%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right) + \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.5%

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

   10. Simplified 88.5%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)} \]

   if 1.5 < x < 2.10000000000000003e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 2.10000000000000003e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;1 + x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{\varepsilon}\right) - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.4% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 300:\\ \;\;\;\;1 + x \cdot \left(\left(0.5 \cdot x\right) \cdot \left(eps\_m \cdot eps\_m\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{eps\_m}\right) - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 300.0)
   (+ 1.0 (* x (* (* 0.5 x) (* eps_m eps_m))))
   (if (<= x 2.1e+115)
     (* x (* 0.25 (* x (* eps_m eps_m))))
     (- (+ 0.5 (/ 0.5 eps_m)) (+ (/ 0.5 eps_m) -0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 300.0) {
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)));
	} else if (x <= 2.1e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	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 <= 300.0d0) then
        tmp = 1.0d0 + (x * ((0.5d0 * x) * (eps_m * eps_m)))
    else if (x <= 2.1d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 + (0.5d0 / eps_m)) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= 300.0) {
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)));
	} else if (x <= 2.1e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 300.0:
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)))
	elif x <= 2.1e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 300.0)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.5 * x) * Float64(eps_m * eps_m))));
	elseif (x <= 2.1e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 + Float64(0.5 / eps_m)) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 300.0)
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)));
	elseif (x <= 2.1e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 + (0.5 / eps_m)) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 300.0], N[(1.0 + N[(x * N[(N[(0.5 * x), $MachinePrecision] * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 300

   1. Initial program 56.8%

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

   3. Simplified 56.8%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   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 57.9%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right) + \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.5%

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

   10. Simplified 88.5%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 88.0%

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

   13. Simplified 88.0%

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

   if 300 < x < 2.10000000000000003e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 2.10000000000000003e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 300:\\ \;\;\;\;1 + x \cdot \left(\left(0.5 \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + \frac{0.5}{\varepsilon}\right) - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 290:\\ \;\;\;\;1 + x \cdot \left(\left(0.5 \cdot x\right) \cdot \left(eps\_m \cdot eps\_m\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{eps\_m} - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 290.0)
   (+ 1.0 (* x (* (* 0.5 x) (* eps_m eps_m))))
   (if (<= x 3.1e+115)
     (* x (* 0.25 (* x (* eps_m eps_m))))
     (- (/ 0.5 eps_m) (+ (/ 0.5 eps_m) -0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 290.0) {
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)));
	} else if (x <= 3.1e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	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 <= 290.0d0) then
        tmp = 1.0d0 + (x * ((0.5d0 * x) * (eps_m * eps_m)))
    else if (x <= 3.1d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 / eps_m) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= 290.0) {
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)));
	} else if (x <= 3.1e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 290.0:
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)))
	elif x <= 3.1e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 290.0)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(0.5 * x) * Float64(eps_m * eps_m))));
	elseif (x <= 3.1e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 / eps_m) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 290.0)
		tmp = 1.0 + (x * ((0.5 * x) * (eps_m * eps_m)));
	elseif (x <= 3.1e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 290.0], N[(1.0 + N[(x * N[(N[(0.5 * x), $MachinePrecision] * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / eps$95$m), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 290

   1. Initial program 56.8%

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

   3. Simplified 56.8%

      \[\leadsto \color{blue}{e^{x \cdot \left(\varepsilon + -1\right)} \cdot \left(0.5 - \frac{-0.5}{\varepsilon}\right) - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)} \]
   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 57.9%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right)\right) + \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right) + 0.5 \cdot \left(\left(\varepsilon + -1\right) \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right) + \left(1 + \varepsilon\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)\right)\right)} \]

   8. Taylor expanded in eps around inf

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

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

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

      2. unpow2 N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 88.5%

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

   10. Simplified 88.5%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \left(0.5 + x \cdot -0.3333333333333333\right)\right)\right)} \]

   11. Taylor expanded in x around 0

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

       1. *-lowering-*.f64 88.0%

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

   13. Simplified 88.0%

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

   if 290 < x < 3.10000000000000005e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 3.10000000000000005e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 67.7%

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

   13. Simplified 67.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 290:\\ \;\;\;\;1 + x \cdot \left(\left(0.5 \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\varepsilon} - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.8% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 300:\\ \;\;\;\;1 + 0.5 \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{eps\_m} - \left(\frac{0.5}{eps\_m} + -0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 300.0)
   (+ 1.0 (* 0.5 (* eps_m (* eps_m (* x x)))))
   (if (<= x 3.4e+115)
     (* x (* 0.25 (* x (* eps_m eps_m))))
     (- (/ 0.5 eps_m) (+ (/ 0.5 eps_m) -0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 300.0) {
		tmp = 1.0 + (0.5 * (eps_m * (eps_m * (x * x))));
	} else if (x <= 3.4e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	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 <= 300.0d0) then
        tmp = 1.0d0 + (0.5d0 * (eps_m * (eps_m * (x * x))))
    else if (x <= 3.4d+115) then
        tmp = x * (0.25d0 * (x * (eps_m * eps_m)))
    else
        tmp = (0.5d0 / eps_m) - ((0.5d0 / eps_m) + (-0.5d0))
    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 <= 300.0) {
		tmp = 1.0 + (0.5 * (eps_m * (eps_m * (x * x))));
	} else if (x <= 3.4e+115) {
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	} else {
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 300.0:
		tmp = 1.0 + (0.5 * (eps_m * (eps_m * (x * x))))
	elif x <= 3.4e+115:
		tmp = x * (0.25 * (x * (eps_m * eps_m)))
	else:
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 300.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(eps_m * Float64(eps_m * Float64(x * x)))));
	elseif (x <= 3.4e+115)
		tmp = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))));
	else
		tmp = Float64(Float64(0.5 / eps_m) - Float64(Float64(0.5 / eps_m) + -0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 300.0)
		tmp = 1.0 + (0.5 * (eps_m * (eps_m * (x * x))));
	elseif (x <= 3.4e+115)
		tmp = x * (0.25 * (x * (eps_m * eps_m)));
	else
		tmp = (0.5 / eps_m) - ((0.5 / eps_m) + -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 300.0], N[(1.0 + N[(0.5 * N[(eps$95$m * N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.4e+115], N[(x * N[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / eps$95$m), $MachinePrecision] - N[(N[(0.5 / eps$95$m), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 300

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 51.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(x \cdot \left(\left(\frac{1}{6} \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \frac{1}{2} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{\varepsilon} - \frac{1}{2}\right)\right)\right)} \]

   10. Simplified 55.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(\left(0.16666666666666666 \cdot x\right) \cdot \left(\left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right) + 0.5 \cdot \left(\left(\frac{0.5}{\varepsilon} + -0.5\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) + \left(\left(\varepsilon + -1\right) \cdot \left(\varepsilon + -1\right)\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right)\right)\right) + \left(\left(\varepsilon + -1\right) \cdot \left(0.5 + \frac{0.5}{\varepsilon}\right) + \left(1 + \varepsilon\right) \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right)\right)\right)} \]

   11. Taylor expanded in eps around inf

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left({\varepsilon}^{3} \cdot \left(\frac{-1}{12} \cdot {x}^{2} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)}\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{3}\right), \color{blue}{\left(\frac{-1}{12} \cdot {x}^{2} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)}\right)\right)\right) \]

       2. cube-mult N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right), \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot {\varepsilon}^{2}\right), \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left({\varepsilon}^{2}\right)\right), \left(\color{blue}{\frac{-1}{12} \cdot {x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \left(\varepsilon \cdot \varepsilon\right)\right), \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \left(\frac{-1}{12} \cdot \color{blue}{{x}^{2}} + \frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left(\frac{-1}{12} \cdot {x}^{2}\right), \color{blue}{\left(\frac{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\left({x}^{2} \cdot \frac{-1}{12}\right), \left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}}{\varepsilon}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \frac{-1}{12}\right), \left(\frac{\color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}}{\varepsilon}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{12}\right), \left(\frac{\color{blue}{x} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \left(\frac{\color{blue}{x} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)}{\varepsilon}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right), \color{blue}{\varepsilon}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot x\right)\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{6}\right)\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 65.5%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \varepsilon\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{12}\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right), \varepsilon\right)\right)\right)\right)\right) \]

   13. Simplified 65.5%

       \[\leadsto 1 + x \cdot \color{blue}{\left(\left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot -0.08333333333333333 + \frac{x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)}{\varepsilon}\right)\right)} \]

   14. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

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

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 86.7%

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

   16. Simplified 86.7%

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

   if 300 < x < 3.4000000000000001e115

   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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 43.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 65.1%

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

   10. Simplified 65.1%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if 3.4000000000000001e115 < 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

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right)}, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \varepsilon\right), \frac{-1}{2}\right)\right)\right) \]
   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 12.5%

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

   7. Simplified 12.5%

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

   8. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 67.7%

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

   10. Simplified 67.7%

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

   11. Taylor expanded in eps around 0

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

       1. /-lowering-/.f64 67.7%

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

   13. Simplified 67.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 300:\\ \;\;\;\;1 + 0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\varepsilon} - \left(\frac{0.5}{\varepsilon} + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.9% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(0.25 \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* 0.25 (* x (* eps_m eps_m))))))
   (if (<= x -4.6e-44) t_0 (if (<= x 0.023) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (0.25 * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -4.6e-44) {
		tmp = t_0;
	} else if (x <= 0.023) {
		tmp = 1.0;
	} else {
		tmp = 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 * (0.25d0 * (x * (eps_m * eps_m)))
    if (x <= (-4.6d-44)) then
        tmp = t_0
    else if (x <= 0.023d0) then
        tmp = 1.0d0
    else
        tmp = 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 * (0.25 * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -4.6e-44) {
		tmp = t_0;
	} else if (x <= 0.023) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (0.25 * (x * (eps_m * eps_m)))
	tmp = 0
	if x <= -4.6e-44:
		tmp = t_0
	elif x <= 0.023:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(0.25 * Float64(x * Float64(eps_m * eps_m))))
	tmp = 0.0
	if (x <= -4.6e-44)
		tmp = t_0;
	elseif (x <= 0.023)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (0.25 * (x * (eps_m * eps_m)));
	tmp = 0.0;
	if (x <= -4.6e-44)
		tmp = t_0;
	elseif (x <= 0.023)
		tmp = 1.0;
	else
		tmp = 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[(0.25 * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e-44], t$95$0, If[LessEqual[x, 0.023], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.59999999999999996e-44 or 0.023 < x

   1. Initial program 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 58.6%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 67.0%

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

   10. Simplified 67.0%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot x} \]

   if -4.59999999999999996e-44 < x < 0.023

   1. Initial program 44.6%

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

   3. Simplified 44.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 83.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-44}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.023:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.25 \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{if}\;x \leq -0.115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.035:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* eps_m (* eps_m (* x x))))))
   (if (<= x -0.115) t_0 (if (<= x 0.035) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.25 * (eps_m * (eps_m * (x * x)));
	double tmp;
	if (x <= -0.115) {
		tmp = t_0;
	} else if (x <= 0.035) {
		tmp = 1.0;
	} else {
		tmp = 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 = 0.25d0 * (eps_m * (eps_m * (x * x)))
    if (x <= (-0.115d0)) then
        tmp = t_0
    else if (x <= 0.035d0) then
        tmp = 1.0d0
    else
        tmp = 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 = 0.25 * (eps_m * (eps_m * (x * x)));
	double tmp;
	if (x <= -0.115) {
		tmp = t_0;
	} else if (x <= 0.035) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 0.25 * (eps_m * (eps_m * (x * x)))
	tmp = 0
	if x <= -0.115:
		tmp = t_0
	elif x <= 0.035:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.25 * Float64(eps_m * Float64(eps_m * Float64(x * x))))
	tmp = 0.0
	if (x <= -0.115)
		tmp = t_0;
	elseif (x <= 0.035)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 0.25 * (eps_m * (eps_m * (x * x)));
	tmp = 0.0;
	if (x <= -0.115)
		tmp = t_0;
	elseif (x <= 0.035)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(0.25 * N[(eps$95$m * N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.115], t$95$0, If[LessEqual[x, 0.035], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.115000000000000005 or 0.035000000000000003 < 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

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

   7. Simplified 59.7%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(x \cdot {\varepsilon}^{2}\right)\right) \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{4} \cdot x\right) \cdot {\varepsilon}^{2}\right) \cdot x \]

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

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 68.9%

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

   10. Simplified 68.9%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

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

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

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

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

       6. *-commutative N/A

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

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

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

       8. *-lowering-*.f64 63.5%

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

   12. Applied egg-rr 63.5%

       \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot 0.25\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)} \]

   13. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 59.8%

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

   15. Simplified 59.8%

       \[\leadsto \color{blue}{0.25 \cdot \left(\left(\varepsilon \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right)} \]

   if -0.115000000000000005 < x < 0.035000000000000003

   1. Initial program 44.8%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 80.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.115:\\ \;\;\;\;0.25 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{elif}\;x \leq 0.035:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.7% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 69.0%

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

3. Simplified 69.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 46.6%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024162 
(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))