NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.7%

Time: 16.1s

Alternatives: 14

Speedup: 14.2×

• 38.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -5.114597943280682e+282
  2. eps = 1.2368870015556245e-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: 73.4% 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: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 126:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - eps\_m\right)} + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 126.0)
   (/ (+ x 1.0) (exp x))
   (/ (+ (exp (* x (- -1.0 eps_m))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * (-1.0 - eps_m))) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 126.0d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (exp((x * ((-1.0d0) - eps_m))) + exp((x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (Math.exp((x * (-1.0 - eps_m))) + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 126.0:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (math.exp((x * (-1.0 - eps_m))) + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 126.0)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 126.0)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (exp((x * (-1.0 - eps_m))) + exp((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 126.0], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 126

   1. Initial program 67.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.4%

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

   5. Simplified 65.4%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 65.4%

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

   7. Applied egg-rr 65.4%

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

   if 126 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 126:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (* x (- -1.0 eps_m)))) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in eps around inf

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

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

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

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

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

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

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-neg-in N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. sub-neg N/A

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

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

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

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

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

5. Simplified 98.6%

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

Alternative 3: 94.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 126:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 + x \cdot \left(\left(-1 - eps\_m\right) + 0.5 \cdot \left(\left(eps\_m + 1\right) \cdot \left(x \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 126.0)
   (/ (+ x 1.0) (exp x))
   (/
    (+
     (exp (* x eps_m))
     (+
      1.0
      (* x (+ (- -1.0 eps_m) (* 0.5 (* (+ eps_m 1.0) (* x (+ eps_m 1.0))))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((eps_m + 1.0) * (x * (eps_m + 1.0)))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 126.0d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (exp((x * eps_m)) + (1.0d0 + (x * (((-1.0d0) - eps_m) + (0.5d0 * ((eps_m + 1.0d0) * (x * (eps_m + 1.0d0)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (Math.exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((eps_m + 1.0) * (x * (eps_m + 1.0)))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 126.0:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (math.exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((eps_m + 1.0) * (x * (eps_m + 1.0)))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 126.0)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 + Float64(x * Float64(Float64(-1.0 - eps_m) + Float64(0.5 * Float64(Float64(eps_m + 1.0) * Float64(x * Float64(eps_m + 1.0)))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 126.0)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((eps_m + 1.0) * (x * (eps_m + 1.0)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 126.0], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(N[(-1.0 - eps$95$m), $MachinePrecision] + N[(0.5 * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 126

   1. Initial program 67.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.4%

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

   5. Simplified 65.4%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 65.4%

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

   7. Applied egg-rr 65.4%

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

   if 126 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

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

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

       6. +-commutative N/A

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. mul-1-neg N/A

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

       11. unsub-neg N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

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

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

   11. Simplified 86.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 126:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(\left(\varepsilon + 1\right) \cdot \left(x \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 126:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \left(1 + x \cdot \left(\left(-1 - eps\_m\right) + 0.5 \cdot \left(\left(x \cdot eps\_m\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 126.0)
   (/ (+ x 1.0) (exp x))
   (/
    (+
     (exp (* x eps_m))
     (+ 1.0 (* x (+ (- -1.0 eps_m) (* 0.5 (* (* x eps_m) (+ eps_m 1.0)))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((x * eps_m) * (eps_m + 1.0))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 126.0d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (exp((x * eps_m)) + (1.0d0 + (x * (((-1.0d0) - eps_m) + (0.5d0 * ((x * eps_m) * (eps_m + 1.0d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (Math.exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((x * eps_m) * (eps_m + 1.0))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 126.0:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (math.exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((x * eps_m) * (eps_m + 1.0))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 126.0)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 + Float64(x * Float64(Float64(-1.0 - eps_m) + Float64(0.5 * Float64(Float64(x * eps_m) * Float64(eps_m + 1.0))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 126.0)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (exp((x * eps_m)) + (1.0 + (x * ((-1.0 - eps_m) + (0.5 * ((x * eps_m) * (eps_m + 1.0))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 126.0], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(N[(-1.0 - eps$95$m), $MachinePrecision] + N[(0.5 * N[(N[(x * eps$95$m), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 126

   1. Initial program 67.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.4%

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

   5. Simplified 65.4%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 65.4%

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

   7. Applied egg-rr 65.4%

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

   if 126 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

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

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

       6. +-commutative N/A

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. mul-1-neg N/A

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

       11. unsub-neg N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

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

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

   11. Simplified 86.9%

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

   12. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 86.9%

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

   14. Simplified 86.9%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 126:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(\left(x \cdot \varepsilon\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{1}{eps\_m}\\ t_1 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;eps\_m \leq 0.027:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;eps\_m \leq 9.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(eps\_m + 1\right) \cdot t\_0 + \left(t\_1 \cdot \left(\left(-1 + eps\_m\right) + \left(x \cdot 0.5\right) \cdot \left(\left(1 - eps\_m\right) \cdot \left(1 - eps\_m\right)\right)\right) + x \cdot \left(t\_0 \cdot \left(-0.5 \cdot \left(\left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{eps\_m} + \left(1 + t\_0\right)\right) \cdot \left(1 + \left(\frac{0}{eps\_m} - -1\right)\right)}{t\_1 + \left(-1 - \frac{-1}{eps\_m}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 1.0 eps_m))) (t_1 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= eps_m 0.027)
     (/ (+ x 1.0) (exp x))
     (if (<= eps_m 9.6e+106)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (+ eps_m 1.0) t_0)
           (+
            (*
             t_1
             (+ (+ -1.0 eps_m) (* (* x 0.5) (* (- 1.0 eps_m) (- 1.0 eps_m)))))
            (* x (* t_0 (* -0.5 (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
        2.0)
       (/
        (/
         (* (+ (/ 1.0 eps_m) (+ 1.0 t_0)) (+ 1.0 (- (/ 0.0 eps_m) -1.0)))
         (+ t_1 (- -1.0 (/ -1.0 eps_m))))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (eps_m <= 0.027) {
		tmp = (x + 1.0) / exp(x);
	} else if (eps_m <= 9.6e+106) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + (1.0d0 / eps_m)
    t_1 = 1.0d0 + (1.0d0 / eps_m)
    if (eps_m <= 0.027d0) then
        tmp = (x + 1.0d0) / exp(x)
    else if (eps_m <= 9.6d+106) then
        tmp = (2.0d0 + (x * (((eps_m + 1.0d0) * t_0) + ((t_1 * (((-1.0d0) + eps_m) + ((x * 0.5d0) * ((1.0d0 - eps_m) * (1.0d0 - eps_m))))) + (x * (t_0 * ((-0.5d0) * ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))) / 2.0d0
    else
        tmp = ((((1.0d0 / eps_m) + (1.0d0 + t_0)) * (1.0d0 + ((0.0d0 / eps_m) - (-1.0d0)))) / (t_1 + ((-1.0d0) - ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = -1.0 + (1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (eps_m <= 0.027) {
		tmp = (x + 1.0) / Math.exp(x);
	} else if (eps_m <= 9.6e+106) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (1.0 / eps_m)
	t_1 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if eps_m <= 0.027:
		tmp = (x + 1.0) / math.exp(x)
	elif eps_m <= 9.6e+106:
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0
	else:
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (eps_m <= 0.027)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	elseif (eps_m <= 9.6e+106)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps_m + 1.0) * t_0) + Float64(Float64(t_1 * Float64(Float64(-1.0 + eps_m) + Float64(Float64(x * 0.5) * Float64(Float64(1.0 - eps_m) * Float64(1.0 - eps_m))))) + Float64(x * Float64(t_0 * Float64(-0.5 * Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))))))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / eps_m) + Float64(1.0 + t_0)) * Float64(1.0 + Float64(Float64(0.0 / eps_m) - -1.0))) / Float64(t_1 + Float64(-1.0 - Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (1.0 / eps_m);
	t_1 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (eps_m <= 0.027)
		tmp = (x + 1.0) / exp(x);
	elseif (eps_m <= 9.6e+106)
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	else
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.027], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 9.6e+106], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(t$95$1 * N[(N[(-1.0 + eps$95$m), $MachinePrecision] + N[(N[(x * 0.5), $MachinePrecision] * N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(t$95$0 * N[(-0.5 * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.0269999999999999997

   1. Initial program 66.9%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.1%

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

   5. Simplified 65.1%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. exp-lowering-exp.f64 65.1%

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

   7. Applied egg-rr 65.1%

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

   if 0.0269999999999999997 < eps < 9.6000000000000002e106

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 80.9%

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

   if 9.6000000000000002e106 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. /-lowering-/.f64 61.6%

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

   5. Simplified 61.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 17.0%

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

   8. Simplified 17.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. sqr-neg N/A

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

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

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

   10. Applied egg-rr 84.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(1 + \frac{-1}{\varepsilon}\right)}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 9.6 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right) + \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(-1 + \varepsilon\right) + \left(x \cdot 0.5\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right)\right) + x \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-0.5 \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 - \frac{-1}{\varepsilon}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 90000:\\ \;\;\;\;e^{0 - x}\\ \mathbf{elif}\;eps\_m \leq 7.1 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(-1 + x \cdot 0.5\right) + eps\_m \cdot \left(x \cdot x + eps\_m \cdot \left(x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{eps\_m} + \left(1 + \left(-1 + \frac{1}{eps\_m}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{eps\_m} - -1\right)\right)}{\left(1 + \frac{1}{eps\_m}\right) + \left(-1 - \frac{-1}{eps\_m}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 90000.0)
   (exp (- 0.0 x))
   (if (<= eps_m 7.1e+106)
     (/
      (+
       2.0
       (+ (* x (+ -1.0 (* x 0.5))) (* eps_m (+ (* x x) (* eps_m (* x x))))))
      2.0)
     (/
      (/
       (*
        (+ (/ 1.0 eps_m) (+ 1.0 (+ -1.0 (/ 1.0 eps_m))))
        (+ 1.0 (- (/ 0.0 eps_m) -1.0)))
       (+ (+ 1.0 (/ 1.0 eps_m)) (- -1.0 (/ -1.0 eps_m))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 90000.0) {
		tmp = exp((0.0 - x));
	} else if (eps_m <= 7.1e+106) {
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 90000.0d0) then
        tmp = exp((0.0d0 - x))
    else if (eps_m <= 7.1d+106) then
        tmp = (2.0d0 + ((x * ((-1.0d0) + (x * 0.5d0))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0d0
    else
        tmp = ((((1.0d0 / eps_m) + (1.0d0 + ((-1.0d0) + (1.0d0 / eps_m)))) * (1.0d0 + ((0.0d0 / eps_m) - (-1.0d0)))) / ((1.0d0 + (1.0d0 / eps_m)) + ((-1.0d0) - ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 90000.0) {
		tmp = Math.exp((0.0 - x));
	} else if (eps_m <= 7.1e+106) {
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 90000.0:
		tmp = math.exp((0.0 - x))
	elif eps_m <= 7.1e+106:
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0
	else:
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 90000.0)
		tmp = exp(Float64(0.0 - x));
	elseif (eps_m <= 7.1e+106)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * Float64(-1.0 + Float64(x * 0.5))) + Float64(eps_m * Float64(Float64(x * x) + Float64(eps_m * Float64(x * x)))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / eps_m) + Float64(1.0 + Float64(-1.0 + Float64(1.0 / eps_m)))) * Float64(1.0 + Float64(Float64(0.0 / eps_m) - -1.0))) / Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(-1.0 - Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 90000.0)
		tmp = exp((0.0 - x));
	elseif (eps_m <= 7.1e+106)
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0;
	else
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 90000.0], N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision], If[LessEqual[eps$95$m, 7.1e+106], N[(N[(2.0 + N[(N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps$95$m * N[(N[(x * x), $MachinePrecision] + N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(1.0 + N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 9e4

   1. Initial program 67.4%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 98.1%

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

   6. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{e^{-1 \cdot x}} \]
   7. Step-by-step derivation

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

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

      2. neg-mul-1 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 75.2%

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

   8. Simplified 75.2%

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

   if 9e4 < eps < 7.1000000000000003e106

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

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

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

       6. +-commutative N/A

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. mul-1-neg N/A

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

       11. unsub-neg N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

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

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

   11. Simplified 82.6%

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

   12. Taylor expanded in eps around 0

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

   14. Simplified 82.6%

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

   if 7.1000000000000003e106 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. /-lowering-/.f64 61.6%

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

   5. Simplified 61.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 17.0%

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

   8. Simplified 17.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. sqr-neg N/A

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

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

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

   10. Applied egg-rr 84.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(1 + \frac{-1}{\varepsilon}\right)}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 90000:\\ \;\;\;\;e^{0 - x}\\ \mathbf{elif}\;\varepsilon \leq 7.1 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(-1 + x \cdot 0.5\right) + \varepsilon \cdot \left(x \cdot x + \varepsilon \cdot \left(x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 - \frac{-1}{\varepsilon}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + \frac{1}{eps\_m}\\ t_1 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;eps\_m \leq 0.027:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;eps\_m \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(eps\_m + 1\right) \cdot t\_0 + \left(t\_1 \cdot \left(\left(-1 + eps\_m\right) + \left(x \cdot 0.5\right) \cdot \left(\left(1 - eps\_m\right) \cdot \left(1 - eps\_m\right)\right)\right) + x \cdot \left(t\_0 \cdot \left(-0.5 \cdot \left(\left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{eps\_m} + \left(1 + t\_0\right)\right) \cdot \left(1 + \left(\frac{0}{eps\_m} - -1\right)\right)}{t\_1 + \left(-1 - \frac{-1}{eps\_m}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (/ 1.0 eps_m))) (t_1 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= eps_m 0.027)
     (/ 1.0 (+ 1.0 (* 0.5 (* x x))))
     (if (<= eps_m 1.1e+107)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (+ eps_m 1.0) t_0)
           (+
            (*
             t_1
             (+ (+ -1.0 eps_m) (* (* x 0.5) (* (- 1.0 eps_m) (- 1.0 eps_m)))))
            (* x (* t_0 (* -0.5 (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
        2.0)
       (/
        (/
         (* (+ (/ 1.0 eps_m) (+ 1.0 t_0)) (+ 1.0 (- (/ 0.0 eps_m) -1.0)))
         (+ t_1 (- -1.0 (/ -1.0 eps_m))))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (eps_m <= 0.027) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else if (eps_m <= 1.1e+107) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) + (1.0d0 / eps_m)
    t_1 = 1.0d0 + (1.0d0 / eps_m)
    if (eps_m <= 0.027d0) then
        tmp = 1.0d0 / (1.0d0 + (0.5d0 * (x * x)))
    else if (eps_m <= 1.1d+107) then
        tmp = (2.0d0 + (x * (((eps_m + 1.0d0) * t_0) + ((t_1 * (((-1.0d0) + eps_m) + ((x * 0.5d0) * ((1.0d0 - eps_m) * (1.0d0 - eps_m))))) + (x * (t_0 * ((-0.5d0) * ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))) / 2.0d0
    else
        tmp = ((((1.0d0 / eps_m) + (1.0d0 + t_0)) * (1.0d0 + ((0.0d0 / eps_m) - (-1.0d0)))) / (t_1 + ((-1.0d0) - ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = -1.0 + (1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (eps_m <= 0.027) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else if (eps_m <= 1.1e+107) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = -1.0 + (1.0 / eps_m)
	t_1 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if eps_m <= 0.027:
		tmp = 1.0 / (1.0 + (0.5 * (x * x)))
	elif eps_m <= 1.1e+107:
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0
	else:
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (eps_m <= 0.027)
		tmp = Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(x * x))));
	elseif (eps_m <= 1.1e+107)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps_m + 1.0) * t_0) + Float64(Float64(t_1 * Float64(Float64(-1.0 + eps_m) + Float64(Float64(x * 0.5) * Float64(Float64(1.0 - eps_m) * Float64(1.0 - eps_m))))) + Float64(x * Float64(t_0 * Float64(-0.5 * Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))))))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / eps_m) + Float64(1.0 + t_0)) * Float64(1.0 + Float64(Float64(0.0 / eps_m) - -1.0))) / Float64(t_1 + Float64(-1.0 - Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = -1.0 + (1.0 / eps_m);
	t_1 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (eps_m <= 0.027)
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	elseif (eps_m <= 1.1e+107)
		tmp = (2.0 + (x * (((eps_m + 1.0) * t_0) + ((t_1 * ((-1.0 + eps_m) + ((x * 0.5) * ((1.0 - eps_m) * (1.0 - eps_m))))) + (x * (t_0 * (-0.5 * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	else
		tmp = ((((1.0 / eps_m) + (1.0 + t_0)) * (1.0 + ((0.0 / eps_m) - -1.0))) / (t_1 + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.027], N[(1.0 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.1e+107], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[(t$95$1 * N[(N[(-1.0 + eps$95$m), $MachinePrecision] + N[(N[(x * 0.5), $MachinePrecision] * N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(t$95$0 * N[(-0.5 * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.0269999999999999997

   1. Initial program 66.9%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.1%

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

   5. Simplified 65.1%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      8. +-lowering-+.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Applied egg-rr 65.1%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   10. Simplified 57.1%

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

   if 0.0269999999999999997 < eps < 1.1e107

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 80.9%

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

   if 1.1e107 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. /-lowering-/.f64 61.6%

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

   5. Simplified 61.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 17.0%

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

   8. Simplified 17.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. sqr-neg N/A

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

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

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

   10. Applied egg-rr 84.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(1 + \frac{-1}{\varepsilon}\right)}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right) + \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(-1 + \varepsilon\right) + \left(x \cdot 0.5\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right)\right) + x \cdot \left(\left(-1 + \frac{1}{\varepsilon}\right) \cdot \left(-0.5 \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 - \frac{-1}{\varepsilon}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 126:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;eps\_m \leq 2.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(-1 + x \cdot 0.5\right) + eps\_m \cdot \left(x \cdot x + eps\_m \cdot \left(x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{eps\_m} + \left(1 + \left(-1 + \frac{1}{eps\_m}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{eps\_m} - -1\right)\right)}{\left(1 + \frac{1}{eps\_m}\right) + \left(-1 - \frac{-1}{eps\_m}\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 126.0)
   (/ 1.0 (+ 1.0 (* 0.5 (* x x))))
   (if (<= eps_m 2.8e+106)
     (/
      (+
       2.0
       (+ (* x (+ -1.0 (* x 0.5))) (* eps_m (+ (* x x) (* eps_m (* x x))))))
      2.0)
     (/
      (/
       (*
        (+ (/ 1.0 eps_m) (+ 1.0 (+ -1.0 (/ 1.0 eps_m))))
        (+ 1.0 (- (/ 0.0 eps_m) -1.0)))
       (+ (+ 1.0 (/ 1.0 eps_m)) (- -1.0 (/ -1.0 eps_m))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else if (eps_m <= 2.8e+106) {
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 126.0d0) then
        tmp = 1.0d0 / (1.0d0 + (0.5d0 * (x * x)))
    else if (eps_m <= 2.8d+106) then
        tmp = (2.0d0 + ((x * ((-1.0d0) + (x * 0.5d0))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0d0
    else
        tmp = ((((1.0d0 / eps_m) + (1.0d0 + ((-1.0d0) + (1.0d0 / eps_m)))) * (1.0d0 + ((0.0d0 / eps_m) - (-1.0d0)))) / ((1.0d0 + (1.0d0 / eps_m)) + ((-1.0d0) - ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else if (eps_m <= 2.8e+106) {
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0;
	} else {
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 126.0:
		tmp = 1.0 / (1.0 + (0.5 * (x * x)))
	elif eps_m <= 2.8e+106:
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0
	else:
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 126.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(x * x))));
	elseif (eps_m <= 2.8e+106)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * Float64(-1.0 + Float64(x * 0.5))) + Float64(eps_m * Float64(Float64(x * x) + Float64(eps_m * Float64(x * x)))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(1.0 / eps_m) + Float64(1.0 + Float64(-1.0 + Float64(1.0 / eps_m)))) * Float64(1.0 + Float64(Float64(0.0 / eps_m) - -1.0))) / Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(-1.0 - Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 126.0)
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	elseif (eps_m <= 2.8e+106)
		tmp = (2.0 + ((x * (-1.0 + (x * 0.5))) + (eps_m * ((x * x) + (eps_m * (x * x)))))) / 2.0;
	else
		tmp = ((((1.0 / eps_m) + (1.0 + (-1.0 + (1.0 / eps_m)))) * (1.0 + ((0.0 / eps_m) - -1.0))) / ((1.0 + (1.0 / eps_m)) + (-1.0 - (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 126.0], N[(1.0 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 2.8e+106], N[(N[(2.0 + N[(N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps$95$m * N[(N[(x * x), $MachinePrecision] + N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(1.0 + N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 126

   1. Initial program 67.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.4%

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

   5. Simplified 65.4%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      8. +-lowering-+.f64 65.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Applied egg-rr 65.4%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.5%

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

   10. Simplified 57.5%

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

   if 126 < eps < 2.79999999999999993e106

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

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

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

       6. +-commutative N/A

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

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

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

       8. distribute-lft-in N/A

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

       9. metadata-eval N/A

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

       10. mul-1-neg N/A

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

       11. unsub-neg N/A

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

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

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

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

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

       16. *-commutative N/A

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

       17. unpow2 N/A

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

       18. associate-*l* N/A

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

       19. *-commutative N/A

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

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

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

   11. Simplified 79.3%

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

   12. Taylor expanded in eps around 0

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

   14. Simplified 79.3%

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

   if 2.79999999999999993e106 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

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

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

      5. /-lowering-/.f64 61.6%

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

   5. Simplified 61.6%

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

   6. Taylor expanded in x around 0

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

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

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

      2. /-lowering-/.f64 17.0%

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

   8. Simplified 17.0%

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

      1. sub-neg N/A

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

      2. flip-+ N/A

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

      3. sqr-neg N/A

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

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

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

   10. Applied egg-rr 84.7%

       \[\leadsto \frac{\color{blue}{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) - \left(1 + \frac{-1}{\varepsilon}\right)}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 126:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;\varepsilon \leq 2.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{2 + \left(x \cdot \left(-1 + x \cdot 0.5\right) + \varepsilon \cdot \left(x \cdot x + \varepsilon \cdot \left(x \cdot x\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{1}{\varepsilon} + \left(1 + \left(-1 + \frac{1}{\varepsilon}\right)\right)\right) \cdot \left(1 + \left(\frac{0}{\varepsilon} - -1\right)\right)}{\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 - \frac{-1}{\varepsilon}\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 126:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(eps\_m + \left(-1 - eps\_m\right)\right) + x \cdot \left(0.5 \cdot \left(eps\_m \cdot eps\_m + \left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 126.0)
   (/ 1.0 (+ 1.0 (* 0.5 (* x x))))
   (/
    (+
     2.0
     (*
      x
      (+
       (+ eps_m (- -1.0 eps_m))
       (* x (* 0.5 (+ (* eps_m eps_m) (* (+ eps_m 1.0) (+ eps_m 1.0))))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else {
		tmp = (2.0 + (x * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0)))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 126.0d0) then
        tmp = 1.0d0 / (1.0d0 + (0.5d0 * (x * x)))
    else
        tmp = (2.0d0 + (x * ((eps_m + ((-1.0d0) - eps_m)) + (x * (0.5d0 * ((eps_m * eps_m) + ((eps_m + 1.0d0) * (eps_m + 1.0d0)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 126.0) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else {
		tmp = (2.0 + (x * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0)))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 126.0:
		tmp = 1.0 / (1.0 + (0.5 * (x * x)))
	else:
		tmp = (2.0 + (x * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0)))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 126.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(x * x))));
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(eps_m + Float64(-1.0 - eps_m)) + Float64(x * Float64(0.5 * Float64(Float64(eps_m * eps_m) + Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0)))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 126.0)
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	else
		tmp = (2.0 + (x * ((eps_m + (-1.0 - eps_m)) + (x * (0.5 * ((eps_m * eps_m) + ((eps_m + 1.0) * (eps_m + 1.0)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 126.0], N[(1.0 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(eps$95$m + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x * N[(0.5 * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] + N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 126

   1. Initial program 67.2%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.4%

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

   5. Simplified 65.4%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      8. +-lowering-+.f64 65.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Applied egg-rr 65.4%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.5%

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

   10. Simplified 57.5%

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

   if 126 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. sub-neg N/A

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

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

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

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

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. associate-+r+ N/A

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

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

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

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

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. mul-1-neg N/A

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

       9. unsub-neg N/A

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

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

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

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

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

       12. distribute-lft-out N/A

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

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

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

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

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

       15. unpow2 N/A

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

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

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

       17. unpow2 N/A

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

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

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

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

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

       20. +-lowering-+.f64 81.1%

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

   11. Simplified 81.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 126:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\varepsilon + \left(-1 - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + \left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.027:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;eps\_m \leq 1.75 \cdot 10^{+120}:\\ \;\;\;\;1 + \frac{x \cdot \left(x \cdot 0.25\right)}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(eps\_m \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.027)
   (/ 1.0 (+ 1.0 (* 0.5 (* x x))))
   (if (<= eps_m 1.75e+120)
     (+ 1.0 (/ (* x (* x 0.25)) eps_m))
     (* eps_m (* eps_m (* (* x x) 0.25))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.027) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else if (eps_m <= 1.75e+120) {
		tmp = 1.0 + ((x * (x * 0.25)) / eps_m);
	} else {
		tmp = eps_m * (eps_m * ((x * x) * 0.25));
	}
	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 <= 0.027d0) then
        tmp = 1.0d0 / (1.0d0 + (0.5d0 * (x * x)))
    else if (eps_m <= 1.75d+120) then
        tmp = 1.0d0 + ((x * (x * 0.25d0)) / eps_m)
    else
        tmp = eps_m * (eps_m * ((x * x) * 0.25d0))
    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 <= 0.027) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else if (eps_m <= 1.75e+120) {
		tmp = 1.0 + ((x * (x * 0.25)) / eps_m);
	} else {
		tmp = eps_m * (eps_m * ((x * x) * 0.25));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.027:
		tmp = 1.0 / (1.0 + (0.5 * (x * x)))
	elif eps_m <= 1.75e+120:
		tmp = 1.0 + ((x * (x * 0.25)) / eps_m)
	else:
		tmp = eps_m * (eps_m * ((x * x) * 0.25))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.027)
		tmp = Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(x * x))));
	elseif (eps_m <= 1.75e+120)
		tmp = Float64(1.0 + Float64(Float64(x * Float64(x * 0.25)) / eps_m));
	else
		tmp = Float64(eps_m * Float64(eps_m * Float64(Float64(x * x) * 0.25)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.027)
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	elseif (eps_m <= 1.75e+120)
		tmp = 1.0 + ((x * (x * 0.25)) / eps_m);
	else
		tmp = eps_m * (eps_m * ((x * x) * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.027], N[(1.0 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.75e+120], N[(1.0 + N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision], N[(eps$95$m * N[(eps$95$m * N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.0269999999999999997

   1. Initial program 66.9%

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

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

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

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

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

      3. metadata-eval N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

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

      8. exp-neg N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. exp-lowering-exp.f64 65.1%

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

   5. Simplified 65.1%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. clear-num N/A

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

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

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

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

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

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

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      8. +-lowering-+.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Applied egg-rr 65.1%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.1%

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

   10. Simplified 57.1%

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

   if 0.0269999999999999997 < eps < 1.75000000000000004e120

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-lft1-in N/A

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

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

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

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

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

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

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

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

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

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

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

      14. sub-neg N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

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

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

      18. /-lowering-/.f64 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 81.4%

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

   9. Taylor expanded in eps around 0

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

       1. associate-*r/ N/A

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

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

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{4} \cdot x\right)\right), \varepsilon\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot x\right)\right), \varepsilon\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{4}\right)\right), \varepsilon\right)\right) \]

       9. *-lowering-*.f64 74.1%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{4}\right)\right), \varepsilon\right)\right) \]

   11. Simplified 74.1%

       \[\leadsto 1 + \color{blue}{\frac{x \cdot \left(x \cdot 0.25\right)}{\varepsilon}} \]

   if 1.75000000000000004e120 < eps

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} + 1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right), 2\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} + e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)}\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 - \varepsilon\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \varepsilon + 1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(-1 \cdot \varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \varepsilon\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \varepsilon\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      19. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right), 2\right) \]

   5. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right), 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right), 2\right) \]

      2. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right), 2\right) \]

   8. Simplified 100.0%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right), 2\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), 2\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot x\right)\right)\right)\right)\right), 2\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right), 2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right), 2\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right), 2\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right), 2\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       8. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       11. unsub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       12. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(\frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot x\right)\right)\right)\right)\right)\right), 2\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(\frac{1}{2} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right)\right), 2\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right)\right), 2\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2} \cdot x\right)\right)\right)\right)\right)\right), 2\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot x\right)\right)\right)\right)\right)\right), 2\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right)\right)\right)\right)\right), 2\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

   11. Simplified 91.0%

       \[\leadsto \frac{e^{x \cdot \varepsilon} + \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)}}{2} \]

   12. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({\varepsilon}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

       2. associate-*r* N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{4}\right)} \]

       3. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{1}{4} \cdot \color{blue}{{x}^{2}}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right) \]

       5. associate-*l* N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{1}{4} \cdot {x}^{2}\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 59.9%

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   14. Simplified 59.9%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.027:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;\varepsilon \leq 1.75 \cdot 10^{+120}:\\ \;\;\;\;1 + \frac{x \cdot \left(x \cdot 0.25\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := eps\_m \cdot \left(eps\_m \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)\\ \mathbf{if}\;x \leq -0.00019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.066:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* eps_m (* eps_m (* (* x x) 0.25)))))
   (if (<= x -0.00019) t_0 (if (<= x 0.066) (/ (- 2.0 (* x x)) 2.0) t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = eps_m * (eps_m * ((x * x) * 0.25));
	double tmp;
	if (x <= -0.00019) {
		tmp = t_0;
	} else if (x <= 0.066) {
		tmp = (2.0 - (x * x)) / 2.0;
	} 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 = eps_m * (eps_m * ((x * x) * 0.25d0))
    if (x <= (-0.00019d0)) then
        tmp = t_0
    else if (x <= 0.066d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    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 = eps_m * (eps_m * ((x * x) * 0.25));
	double tmp;
	if (x <= -0.00019) {
		tmp = t_0;
	} else if (x <= 0.066) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = eps_m * (eps_m * ((x * x) * 0.25))
	tmp = 0
	if x <= -0.00019:
		tmp = t_0
	elif x <= 0.066:
		tmp = (2.0 - (x * x)) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(eps_m * Float64(eps_m * Float64(Float64(x * x) * 0.25)))
	tmp = 0.0
	if (x <= -0.00019)
		tmp = t_0;
	elseif (x <= 0.066)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = eps_m * (eps_m * ((x * x) * 0.25));
	tmp = 0.0;
	if (x <= -0.00019)
		tmp = t_0;
	elseif (x <= 0.066)
		tmp = (2.0 - (x * x)) / 2.0;
	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[(eps$95$m * N[(eps$95$m * N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00019], t$95$0, If[LessEqual[x, 0.066], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9000000000000001e-4 or 0.066000000000000003 < x

   1. Initial program 99.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}, 2\right) \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right) \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right), 2\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} + 1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right), 2\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} + e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right), 2\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)}\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 - \varepsilon\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + -1 \cdot \varepsilon\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \varepsilon + 1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(-1 \cdot \varepsilon\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \left(\varepsilon - 1\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon - 1\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(\varepsilon + -1\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \varepsilon\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \varepsilon\right)\right)\right), \left(e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)\right), 2\right) \]

      19. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \varepsilon\right)\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right), 2\right) \]

   5. Simplified 99.1%

      \[\leadsto \frac{\color{blue}{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\varepsilon \cdot x\right)}\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right), 2\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\left(x \cdot \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right), 2\right) \]

      2. *-lowering-*.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right)\right)\right), 2\right) \]

   8. Simplified 75.4%

      \[\leadsto \frac{e^{\color{blue}{x \cdot \varepsilon}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \color{blue}{\left(1 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)}\right), 2\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \frac{1}{2} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right), 2\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot x\right)\right)\right)\right)\right), 2\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right), 2\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right), 2\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x + -1 \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right), 2\right) \]

       6. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 \cdot \left(1 + \varepsilon\right) + \left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right), 2\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot \left(1 + \varepsilon\right)\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       8. distribute-lft-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 \cdot 1 + -1 \cdot \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       9. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + -1 \cdot \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       10. mul-1-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       11. unsub-neg N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(-1 - \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       12. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(\left(\frac{1}{2} \cdot {\left(1 + \varepsilon\right)}^{2}\right) \cdot x\right)\right)\right)\right)\right), 2\right) \]

       13. associate-*r* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(\frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot x\right)\right)\right)\right)\right)\right), 2\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \left(\frac{1}{2} \cdot \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right)\right), 2\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot {\left(1 + \varepsilon\right)}^{2}\right)\right)\right)\right)\right)\right), 2\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left({\left(1 + \varepsilon\right)}^{2} \cdot x\right)\right)\right)\right)\right)\right), 2\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right) \cdot x\right)\right)\right)\right)\right)\right), 2\right) \]

       18. associate-*l* N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)\right)\right)\right)\right)\right)\right), 2\right) \]

       19. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \left(\left(1 + \varepsilon\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(x, \varepsilon\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{\_.f64}\left(-1, \varepsilon\right), \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(1 + \varepsilon\right), \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)\right)\right)\right), 2\right) \]

   11. Simplified 62.5%

       \[\leadsto \frac{e^{x \cdot \varepsilon} + \color{blue}{\left(1 + x \cdot \left(\left(-1 - \varepsilon\right) + 0.5 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)\right)\right)}}{2} \]

   12. Taylor expanded in eps around inf

       \[\leadsto \color{blue}{\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({\varepsilon}^{2} \cdot {x}^{2}\right) \cdot \color{blue}{\frac{1}{4}} \]

       2. associate-*r* N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{1}{4}\right)} \]

       3. *-commutative N/A

          \[\leadsto {\varepsilon}^{2} \cdot \left(\frac{1}{4} \cdot \color{blue}{{x}^{2}}\right) \]

       4. unpow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{4}} \cdot {x}^{2}\right) \]

       5. associate-*l* N/A

          \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2}\right)\right)}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{1}{4} \cdot {x}^{2}\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{1}{4}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 64.2%

           \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   14. Simplified 64.2%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(0.25 \cdot \left(x \cdot x\right)\right)\right)} \]

   if -1.9000000000000001e-4 < x < 0.066000000000000003

   1. Initial program 57.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - -1 \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) + 1 \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + 1\right) \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(x + 1\right) \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(x + 1\right) \cdot \frac{1}{e^{x}}\right)\right), 2\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\left(x + 1\right) \cdot 1}{e^{x}}\right)\right), 2\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{x + 1}{e^{x}}\right)\right), 2\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(x + 1\right), \left(e^{x}\right)\right)\right), 2\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{x}\right)\right)\right), 2\right) \]

      13. exp-lowering-exp.f64 72.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(x\right)\right)\right), 2\right) \]

   5. Simplified 72.4%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{x + 1}{e^{x}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(2 + -1 \cdot {x}^{2}\right)}, 2\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 + \left(\mathsf{neg}\left({x}^{2}\right)\right)\right), 2\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 - {x}^{2}\right), 2\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(2, \left({x}^{2}\right)\right), 2\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(2, \left(x \cdot x\right)\right), 2\right) \]

      5. *-lowering-*.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right), 2\right) \]

   8. Simplified 71.9%

      \[\leadsto \frac{\color{blue}{2 - x \cdot x}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00019:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)\\ \mathbf{elif}\;x \leq 0.066:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.027:\\ \;\;\;\;\frac{1}{1 + 0.5 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(\left(eps\_m \cdot eps\_m\right) \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.027)
   (/ 1.0 (+ 1.0 (* 0.5 (* x x))))
   (+ 1.0 (* x (* x (* (* eps_m eps_m) 0.25))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.027) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else {
		tmp = 1.0 + (x * (x * ((eps_m * eps_m) * 0.25)));
	}
	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 <= 0.027d0) then
        tmp = 1.0d0 / (1.0d0 + (0.5d0 * (x * x)))
    else
        tmp = 1.0d0 + (x * (x * ((eps_m * eps_m) * 0.25d0)))
    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 <= 0.027) {
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	} else {
		tmp = 1.0 + (x * (x * ((eps_m * eps_m) * 0.25)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.027:
		tmp = 1.0 / (1.0 + (0.5 * (x * x)))
	else:
		tmp = 1.0 + (x * (x * ((eps_m * eps_m) * 0.25)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.027)
		tmp = Float64(1.0 / Float64(1.0 + Float64(0.5 * Float64(x * x))));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * Float64(Float64(eps_m * eps_m) * 0.25))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.027)
		tmp = 1.0 / (1.0 + (0.5 * (x * x)));
	else
		tmp = 1.0 + (x * (x * ((eps_m * eps_m) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.027], N[(1.0 / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0269999999999999997

   1. Initial program 66.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)}, 2\right) \]
   4. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - -1 \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) + 1 \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      4. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + 1\right) \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(x + 1\right) \cdot e^{\mathsf{neg}\left(x\right)}\right)\right), 2\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\left(x + 1\right) \cdot \frac{1}{e^{x}}\right)\right), 2\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{\left(x + 1\right) \cdot 1}{e^{x}}\right)\right), 2\right) \]

      10. *-rgt-identity N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{x + 1}{e^{x}}\right)\right), 2\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(x + 1\right), \left(e^{x}\right)\right)\right), 2\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \left(e^{x}\right)\right)\right), 2\right) \]

      13. exp-lowering-exp.f64 65.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), \mathsf{exp.f64}\left(x\right)\right)\right), 2\right) \]

   5. Simplified 65.1%

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{x + 1}{e^{x}}}}{2} \]
   6. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{2 \cdot \frac{x + 1}{e^{x}}}}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{1}{\frac{\frac{2}{2}}{\color{blue}{\frac{x + 1}{e^{x}}}}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{1}{\frac{\color{blue}{x + 1}}{e^{x}}}} \]

      4. clear-num N/A

         \[\leadsto \frac{1}{\frac{e^{x}}{\color{blue}{x + 1}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{e^{x}}{x + 1}\right)}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(e^{x}\right), \color{blue}{\left(x + 1\right)}\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \left(\color{blue}{x} + 1\right)\right)\right) \]

      8. +-lowering-+.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(x\right), \mathsf{+.f64}\left(x, \color{blue}{1}\right)\right)\right) \]

   7. Applied egg-rr 65.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x}}{x + 1}}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2}\right)}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \left(x \cdot \color{blue}{x}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 57.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right)\right) \]

   10. Simplified 57.1%

       \[\leadsto \frac{1}{\color{blue}{1 + 0.5 \cdot \left(x \cdot x\right)}} \]

   if 0.0269999999999999997 < eps

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \color{blue}{\left(\left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\varepsilon}\right) - 1\right)}\right), 2\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(\frac{1}{\varepsilon} - 1\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right) + 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right) + 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 2\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} + -1\right)\right)\right), 2\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(-1 + \frac{1}{\varepsilon}\right)\right)\right), 2\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \mathsf{+.f64}\left(-1, \left(\frac{1}{\varepsilon}\right)\right)\right)\right), 2\right) \]

      18. /-lowering-/.f64 63.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(1, \varepsilon\right)\right)\right)\right), 2\right) \]

   5. Simplified 63.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(x \cdot \left(-1 - \varepsilon\right) + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{4} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right)\right) + \frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   8. Simplified 80.5%

      \[\leadsto \color{blue}{1 + x \cdot \left(\left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right) + \left(0.5 + 0.5 \cdot \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + -1\right)\right) + 0.25 \cdot \left(\left(\left(x \cdot \left(\varepsilon + -1\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon + -1\right)\right)\right)} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{4} \cdot \left({\varepsilon}^{2} \cdot x\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{4} \cdot {\varepsilon}^{2}\right) \cdot \color{blue}{x}\right)\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot {\varepsilon}^{2}\right) \cdot x\right)\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right) \cdot x\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \left(\frac{1}{2} \cdot {\varepsilon}^{2}\right)\right)}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{2}\right) \cdot \color{blue}{{\varepsilon}^{2}}\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{4} \cdot {\color{blue}{\varepsilon}}^{2}\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({\varepsilon}^{2} \cdot \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left({\varepsilon}^{2}\right), \color{blue}{\frac{1}{4}}\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\left(\varepsilon \cdot \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

       11. *-lowering-*.f64 81.3%

           \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\varepsilon, \varepsilon\right), \frac{1}{4}\right)\right)\right)\right) \]

   11. Simplified 81.3%

       \[\leadsto 1 + x \cdot \color{blue}{\left(x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.7% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00019:\\ \;\;\;\;eps\_m \cdot \left(x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.00019) (* eps_m (* x -0.5)) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.00019) {
		tmp = eps_m * (x * -0.5);
	} else {
		tmp = 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) :: tmp
    if (x <= (-0.00019d0)) then
        tmp = eps_m * (x * (-0.5d0))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.00019) {
		tmp = eps_m * (x * -0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.00019:
		tmp = eps_m * (x * -0.5)
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.00019)
		tmp = Float64(eps_m * Float64(x * -0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.00019)
		tmp = eps_m * (x * -0.5);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.00019], N[(eps$95$m * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.9000000000000001e-4

   1. Initial program 97.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \color{blue}{\left(\left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\varepsilon}\right) - 1\right)}\right), 2\right) \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(-1 \cdot \left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right) \cdot \left(\frac{1}{\varepsilon} - 1\right) + \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      5. distribute-lft1-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \left(\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right) + 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right) + 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      8. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(1 + \varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      10. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(1\right)\right) + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(-1 + \left(\mathsf{neg}\left(\varepsilon\right)\right)\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(-1 - \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} - 1\right)\right)\right), 2\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 2\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(\frac{1}{\varepsilon} + -1\right)\right)\right), 2\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \left(-1 + \frac{1}{\varepsilon}\right)\right)\right), 2\right) \]

      17. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \mathsf{+.f64}\left(-1, \left(\frac{1}{\varepsilon}\right)\right)\right)\right), 2\right) \]

      18. /-lowering-/.f64 53.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(1, \varepsilon\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \varepsilon\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-1, \varepsilon\right)\right), 1\right), \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(1, \varepsilon\right)\right)\right)\right), 2\right) \]

   5. Simplified 53.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(x \cdot \left(-1 - \varepsilon\right) + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon \cdot x\right) \cdot \color{blue}{\frac{-1}{2}} \]

      2. associate-*r* N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \frac{-1}{2}\right)} \]

      3. *-commutative N/A

         \[\leadsto \varepsilon \cdot \left(\frac{-1}{2} \cdot \color{blue}{x}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \color{blue}{\left(\frac{-1}{2} \cdot x\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \left(x \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

      6. *-lowering-*.f64 19.7%

         \[\leadsto \mathsf{*.f64}\left(\varepsilon, \mathsf{*.f64}\left(x, \color{blue}{\frac{-1}{2}}\right)\right) \]

   8. Simplified 19.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot -0.5\right)} \]

   if -1.9000000000000001e-4 < x

   1. Initial program 71.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Simplified 50.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 43.5% 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 75.0%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 43.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))