NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 97.7%

Time: 18.9s

Alternatives: 13

Speedup: 14.2×

• 37.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.5078443533838272e+122
  2. eps = -1.3768336755767094e-151

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: 72.7% 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: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(1 - eps\_m\right)}\\ t_1 := e^{x \cdot \left(-1 - eps\_m\right)}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-228}:\\ \;\;\;\;0.5 + 0.5 \cdot t\_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;0.5 + \frac{0.5}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{eps\_m}}{t\_0} + t\_1 \cdot \left(0.5 - \frac{0.5}{eps\_m}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- 1.0 eps_m)))) (t_1 (exp (* x (- -1.0 eps_m)))))
   (if (<= x -9.5e-228)
     (+ 0.5 (* 0.5 t_1))
     (if (<= x 7e-10)
       (+ 0.5 (/ 0.5 t_0))
       (+ (/ (+ 0.5 (/ 0.5 eps_m)) t_0) (* t_1 (- 0.5 (/ 0.5 eps_m))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (1.0 - eps_m)));
	double t_1 = exp((x * (-1.0 - eps_m)));
	double tmp;
	if (x <= -9.5e-228) {
		tmp = 0.5 + (0.5 * t_1);
	} else if (x <= 7e-10) {
		tmp = 0.5 + (0.5 / t_0);
	} else {
		tmp = ((0.5 + (0.5 / eps_m)) / t_0) + (t_1 * (0.5 - (0.5 / eps_m)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((x * (1.0d0 - eps_m)))
    t_1 = exp((x * ((-1.0d0) - eps_m)))
    if (x <= (-9.5d-228)) then
        tmp = 0.5d0 + (0.5d0 * t_1)
    else if (x <= 7d-10) then
        tmp = 0.5d0 + (0.5d0 / t_0)
    else
        tmp = ((0.5d0 + (0.5d0 / eps_m)) / t_0) + (t_1 * (0.5d0 - (0.5d0 / eps_m)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (1.0 - eps_m)));
	double t_1 = Math.exp((x * (-1.0 - eps_m)));
	double tmp;
	if (x <= -9.5e-228) {
		tmp = 0.5 + (0.5 * t_1);
	} else if (x <= 7e-10) {
		tmp = 0.5 + (0.5 / t_0);
	} else {
		tmp = ((0.5 + (0.5 / eps_m)) / t_0) + (t_1 * (0.5 - (0.5 / eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (1.0 - eps_m)))
	t_1 = math.exp((x * (-1.0 - eps_m)))
	tmp = 0
	if x <= -9.5e-228:
		tmp = 0.5 + (0.5 * t_1)
	elif x <= 7e-10:
		tmp = 0.5 + (0.5 / t_0)
	else:
		tmp = ((0.5 + (0.5 / eps_m)) / t_0) + (t_1 * (0.5 - (0.5 / eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(1.0 - eps_m)))
	t_1 = exp(Float64(x * Float64(-1.0 - eps_m)))
	tmp = 0.0
	if (x <= -9.5e-228)
		tmp = Float64(0.5 + Float64(0.5 * t_1));
	elseif (x <= 7e-10)
		tmp = Float64(0.5 + Float64(0.5 / t_0));
	else
		tmp = Float64(Float64(Float64(0.5 + Float64(0.5 / eps_m)) / t_0) + Float64(t_1 * Float64(0.5 - Float64(0.5 / eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (1.0 - eps_m)));
	t_1 = exp((x * (-1.0 - eps_m)));
	tmp = 0.0;
	if (x <= -9.5e-228)
		tmp = 0.5 + (0.5 * t_1);
	elseif (x <= 7e-10)
		tmp = 0.5 + (0.5 / t_0);
	else
		tmp = ((0.5 + (0.5 / eps_m)) / t_0) + (t_1 * (0.5 - (0.5 / eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -9.5e-228], N[(0.5 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-10], N[(0.5 + N[(0.5 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(t$95$1 * N[(0.5 - N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.50000000000000024e-228

   1. Initial program 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

   10. Simplified 67.8%

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

   if -9.50000000000000024e-228 < x < 6.99999999999999961e-10

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 48.1%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

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

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

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

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

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

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

      13. --lowering--.f64 94.2%

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

   10. Simplified 94.2%

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

   if 6.99999999999999961e-10 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-228}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.5}{\varepsilon}}{e^{x \cdot \left(1 - \varepsilon\right)}} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(0.5 - \frac{0.5}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-228}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - eps\_m\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+58}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x \cdot \left(1 - eps\_m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -9.5e-228)
   (+ 0.5 (* 0.5 (exp (* x (- -1.0 eps_m)))))
   (if (<= x 1.95e+58)
     (+ 0.5 (/ 0.5 (exp (* x (- 1.0 eps_m)))))
     (/ (* x (* x (* eps_m eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.5e-228) {
		tmp = 0.5 + (0.5 * exp((x * (-1.0 - eps_m))));
	} else if (x <= 1.95e+58) {
		tmp = 0.5 + (0.5 / exp((x * (1.0 - eps_m))));
	} else {
		tmp = (x * (x * (eps_m * 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 (x <= (-9.5d-228)) then
        tmp = 0.5d0 + (0.5d0 * exp((x * ((-1.0d0) - eps_m))))
    else if (x <= 1.95d+58) then
        tmp = 0.5d0 + (0.5d0 / exp((x * (1.0d0 - eps_m))))
    else
        tmp = (x * (x * (eps_m * 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 (x <= -9.5e-228) {
		tmp = 0.5 + (0.5 * Math.exp((x * (-1.0 - eps_m))));
	} else if (x <= 1.95e+58) {
		tmp = 0.5 + (0.5 / Math.exp((x * (1.0 - eps_m))));
	} else {
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -9.5e-228:
		tmp = 0.5 + (0.5 * math.exp((x * (-1.0 - eps_m))))
	elif x <= 1.95e+58:
		tmp = 0.5 + (0.5 / math.exp((x * (1.0 - eps_m))))
	else:
		tmp = (x * (x * (eps_m * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -9.5e-228)
		tmp = Float64(0.5 + Float64(0.5 * exp(Float64(x * Float64(-1.0 - eps_m)))));
	elseif (x <= 1.95e+58)
		tmp = Float64(0.5 + Float64(0.5 / exp(Float64(x * Float64(1.0 - eps_m)))));
	else
		tmp = Float64(Float64(x * Float64(x * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -9.5e-228)
		tmp = 0.5 + (0.5 * exp((x * (-1.0 - eps_m))));
	elseif (x <= 1.95e+58)
		tmp = 0.5 + (0.5 / exp((x * (1.0 - eps_m))));
	else
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -9.5e-228], N[(0.5 + N[(0.5 * N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e+58], N[(0.5 + N[(0.5 / N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.50000000000000024e-228

   1. Initial program 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

   10. Simplified 67.8%

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

   if -9.50000000000000024e-228 < x < 1.95000000000000005e58

   1. Initial program 60.5%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 44.4%

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

   8. Taylor expanded in eps around inf

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

      1. associate-*r/ N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r/ N/A

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

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

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

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

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

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

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

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

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

      13. --lowering--.f64 83.8%

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

   10. Simplified 83.8%

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

   if 1.95000000000000005e58 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 58.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 57.8%

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

   8. Simplified 57.8%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 57.6%

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

   11. Simplified 57.6%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right), 2\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), 2\right) \]

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

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

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

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

       6. *-lowering-*.f64 80.4%

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

   13. Applied egg-rr 80.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-228}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+58}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x \cdot \left(1 - \varepsilon\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-256}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - eps\_m\right)}\\ \mathbf{elif}\;x \leq 285:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + eps\_m\right) \cdot \left(1 + \frac{1}{eps\_m}\right) + \left(\frac{1}{eps\_m} + x \cdot \left(\left(0.5 + \frac{0.5}{eps\_m}\right) \cdot \left(\left(1 - eps\_m\right) \cdot \left(1 - eps\_m\right)\right) + \left(-0.5 \cdot \left(-1 + \frac{1}{eps\_m}\right)\right) \cdot \left(\left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 7.8e-256)
   (+ 0.5 (* 0.5 (exp (* x (- -1.0 eps_m)))))
   (if (<= x 285.0)
     (/
      (+
       2.0
       (*
        x
        (+
         (* (+ -1.0 eps_m) (+ 1.0 (/ 1.0 eps_m)))
         (+
          (/ 1.0 eps_m)
          (*
           x
           (+
            (* (+ 0.5 (/ 0.5 eps_m)) (* (- 1.0 eps_m) (- 1.0 eps_m)))
            (*
             (* -0.5 (+ -1.0 (/ 1.0 eps_m)))
             (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
      2.0)
     (/ (* x (* x (* eps_m eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 7.8e-256) {
		tmp = 0.5 + (0.5 * exp((x * (-1.0 - eps_m))));
	} else if (x <= 285.0) {
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = (x * (x * (eps_m * 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 (x <= 7.8d-256) then
        tmp = 0.5d0 + (0.5d0 * exp((x * ((-1.0d0) - eps_m))))
    else if (x <= 285.0d0) then
        tmp = (2.0d0 + (x * ((((-1.0d0) + eps_m) * (1.0d0 + (1.0d0 / eps_m))) + ((1.0d0 / eps_m) + (x * (((0.5d0 + (0.5d0 / eps_m)) * ((1.0d0 - eps_m) * (1.0d0 - eps_m))) + (((-0.5d0) * ((-1.0d0) + (1.0d0 / eps_m))) * ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))) / 2.0d0
    else
        tmp = (x * (x * (eps_m * 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 (x <= 7.8e-256) {
		tmp = 0.5 + (0.5 * Math.exp((x * (-1.0 - eps_m))));
	} else if (x <= 285.0) {
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 7.8e-256:
		tmp = 0.5 + (0.5 * math.exp((x * (-1.0 - eps_m))))
	elif x <= 285.0:
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0
	else:
		tmp = (x * (x * (eps_m * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 7.8e-256)
		tmp = Float64(0.5 + Float64(0.5 * exp(Float64(x * Float64(-1.0 - eps_m)))));
	elseif (x <= 285.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 + eps_m) * Float64(1.0 + Float64(1.0 / eps_m))) + Float64(Float64(1.0 / eps_m) + Float64(x * Float64(Float64(Float64(0.5 + Float64(0.5 / eps_m)) * Float64(Float64(1.0 - eps_m) * Float64(1.0 - eps_m))) + Float64(Float64(-0.5 * Float64(-1.0 + Float64(1.0 / eps_m))) * Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))))))))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(x * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 7.8e-256)
		tmp = 0.5 + (0.5 * exp((x * (-1.0 - eps_m))));
	elseif (x <= 285.0)
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	else
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 7.8e-256], N[(0.5 + N[(0.5 * N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 285.0], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * N[(N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.7999999999999997e-256

   1. Initial program 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 46.0%

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

   7. Simplified 46.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

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

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. +-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

   10. Simplified 76.1%

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

   if 7.7999999999999997e-256 < x < 285

   1. Initial program 50.7%

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

   3. Taylor expanded in 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 91.5%

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

   6. Taylor expanded in eps around 0

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

      1. /-lowering-/.f64 91.5%

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

   8. Simplified 91.5%

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

   if 285 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 54.3%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.1%

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

   8. Simplified 54.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 55.2%

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

   11. Simplified 55.2%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right), 2\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), 2\right) \]

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

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

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

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

       6. *-lowering-*.f64 73.7%

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

   13. Applied egg-rr 73.7%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{-256}:\\ \;\;\;\;0.5 + 0.5 \cdot e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{elif}\;x \leq 285:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1}{\varepsilon} + x \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right) + \left(-0.5 \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(\frac{1}{eps\_m} \cdot \left(eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-256}:\\ \;\;\;\;\frac{2 + \left(x \cdot eps\_m\right) \cdot \left(x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 285:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + eps\_m\right) \cdot \left(1 + \frac{1}{eps\_m}\right) + \left(\frac{1}{eps\_m} + x \cdot \left(\left(0.5 + \frac{0.5}{eps\_m}\right) \cdot \left(\left(1 - eps\_m\right) \cdot \left(1 - eps\_m\right)\right) + \left(-0.5 \cdot \left(-1 + \frac{1}{eps\_m}\right)\right) \cdot \left(\left(eps\_m + 1\right) \cdot \left(eps\_m + 1\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.6e-206)
   (/ (+ 2.0 (* x (* x (* (/ 1.0 eps_m) (* eps_m (* eps_m eps_m)))))) 2.0)
   (if (<= x 8e-256)
     (/ (+ 2.0 (* (* x eps_m) (* x eps_m))) 2.0)
     (if (<= x 285.0)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (+ -1.0 eps_m) (+ 1.0 (/ 1.0 eps_m)))
           (+
            (/ 1.0 eps_m)
            (*
             x
             (+
              (* (+ 0.5 (/ 0.5 eps_m)) (* (- 1.0 eps_m) (- 1.0 eps_m)))
              (*
               (* -0.5 (+ -1.0 (/ 1.0 eps_m)))
               (* (+ eps_m 1.0) (+ eps_m 1.0)))))))))
        2.0)
       (/ (* x (* x (* eps_m eps_m))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.6e-206) {
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0;
	} else if (x <= 8e-256) {
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0;
	} else if (x <= 285.0) {
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = (x * (x * (eps_m * 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 (x <= (-2.6d-206)) then
        tmp = (2.0d0 + (x * (x * ((1.0d0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0d0
    else if (x <= 8d-256) then
        tmp = (2.0d0 + ((x * eps_m) * (x * eps_m))) / 2.0d0
    else if (x <= 285.0d0) then
        tmp = (2.0d0 + (x * ((((-1.0d0) + eps_m) * (1.0d0 + (1.0d0 / eps_m))) + ((1.0d0 / eps_m) + (x * (((0.5d0 + (0.5d0 / eps_m)) * ((1.0d0 - eps_m) * (1.0d0 - eps_m))) + (((-0.5d0) * ((-1.0d0) + (1.0d0 / eps_m))) * ((eps_m + 1.0d0) * (eps_m + 1.0d0))))))))) / 2.0d0
    else
        tmp = (x * (x * (eps_m * 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 (x <= -2.6e-206) {
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0;
	} else if (x <= 8e-256) {
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0;
	} else if (x <= 285.0) {
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	} else {
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.6e-206:
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0
	elif x <= 8e-256:
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0
	elif x <= 285.0:
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0
	else:
		tmp = (x * (x * (eps_m * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.6e-206)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x * Float64(Float64(1.0 / eps_m) * Float64(eps_m * Float64(eps_m * eps_m)))))) / 2.0);
	elseif (x <= 8e-256)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * eps_m) * Float64(x * eps_m))) / 2.0);
	elseif (x <= 285.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 + eps_m) * Float64(1.0 + Float64(1.0 / eps_m))) + Float64(Float64(1.0 / eps_m) + Float64(x * Float64(Float64(Float64(0.5 + Float64(0.5 / eps_m)) * Float64(Float64(1.0 - eps_m) * Float64(1.0 - eps_m))) + Float64(Float64(-0.5 * Float64(-1.0 + Float64(1.0 / eps_m))) * Float64(Float64(eps_m + 1.0) * Float64(eps_m + 1.0))))))))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(x * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.6e-206)
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0;
	elseif (x <= 8e-256)
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0;
	elseif (x <= 285.0)
		tmp = (2.0 + (x * (((-1.0 + eps_m) * (1.0 + (1.0 / eps_m))) + ((1.0 / eps_m) + (x * (((0.5 + (0.5 / eps_m)) * ((1.0 - eps_m) * (1.0 - eps_m))) + ((-0.5 * (-1.0 + (1.0 / eps_m))) * ((eps_m + 1.0) * (eps_m + 1.0))))))))) / 2.0;
	else
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.6e-206], N[(N[(2.0 + N[(x * N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] * N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8e-256], N[(N[(2.0 + N[(N[(x * eps$95$m), $MachinePrecision] * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 285.0], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * N[(N[(N[(0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6e-206

   1. Initial program 72.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 92.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.6%

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

   8. Simplified 92.6%

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

      1. pow2 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. inv-pow N/A

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

      5. cube-unmult N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.8%

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

   10. Applied egg-rr 96.8%

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

   if -2.6e-206 < x < 7.99999999999999982e-256

   1. Initial program 64.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 83.4%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 83.4%

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

   8. Simplified 83.4%

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

      1. associate-*r* N/A

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

      2. unswap-sqr N/A

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

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

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

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

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

      5. *-lowering-*.f64 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 7.99999999999999982e-256 < x < 285

   1. Initial program 50.7%

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

   3. Taylor expanded in 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 91.5%

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

   6. Taylor expanded in eps around 0

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

      1. /-lowering-/.f64 91.5%

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

   8. Simplified 91.5%

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

   if 285 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 54.3%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.1%

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

   8. Simplified 54.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 55.2%

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

   11. Simplified 55.2%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right), 2\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), 2\right) \]

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

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

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

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

       6. *-lowering-*.f64 73.7%

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

   13. Applied egg-rr 73.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-206}:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(\frac{1}{\varepsilon} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-256}:\\ \;\;\;\;\frac{2 + \left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 285:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \left(\frac{1}{\varepsilon} + x \cdot \left(\left(0.5 + \frac{0.5}{\varepsilon}\right) \cdot \left(\left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right)\right) + \left(-0.5 \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\left(\varepsilon + 1\right) \cdot \left(\varepsilon + 1\right)\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(\frac{1}{eps\_m} \cdot \left(eps\_m \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-255}:\\ \;\;\;\;\frac{2 + \left(x \cdot eps\_m\right) \cdot \left(x \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 335:\\ \;\;\;\;\frac{2 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps_m eps_m)))))
   (if (<= x -4.5e-207)
     (/ (+ 2.0 (* x (* x (* (/ 1.0 eps_m) (* eps_m (* eps_m eps_m)))))) 2.0)
     (if (<= x 7e-255)
       (/ (+ 2.0 (* (* x eps_m) (* x eps_m))) 2.0)
       (if (<= x 335.0) (/ (+ 2.0 t_0) 2.0) (/ t_0 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x * (eps_m * eps_m));
	double tmp;
	if (x <= -4.5e-207) {
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0;
	} else if (x <= 7e-255) {
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0;
	} else if (x <= 335.0) {
		tmp = (2.0 + t_0) / 2.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (eps_m * eps_m))
    if (x <= (-4.5d-207)) then
        tmp = (2.0d0 + (x * (x * ((1.0d0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0d0
    else if (x <= 7d-255) then
        tmp = (2.0d0 + ((x * eps_m) * (x * eps_m))) / 2.0d0
    else if (x <= 335.0d0) then
        tmp = (2.0d0 + t_0) / 2.0d0
    else
        tmp = t_0 / 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 = x * (x * (eps_m * eps_m));
	double tmp;
	if (x <= -4.5e-207) {
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0;
	} else if (x <= 7e-255) {
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0;
	} else if (x <= 335.0) {
		tmp = (2.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (x * (eps_m * eps_m))
	tmp = 0
	if x <= -4.5e-207:
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0
	elif x <= 7e-255:
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0
	elif x <= 335.0:
		tmp = (2.0 + t_0) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x * Float64(eps_m * eps_m)))
	tmp = 0.0
	if (x <= -4.5e-207)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x * Float64(Float64(1.0 / eps_m) * Float64(eps_m * Float64(eps_m * eps_m)))))) / 2.0);
	elseif (x <= 7e-255)
		tmp = Float64(Float64(2.0 + Float64(Float64(x * eps_m) * Float64(x * eps_m))) / 2.0);
	elseif (x <= 335.0)
		tmp = Float64(Float64(2.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (x * (eps_m * eps_m));
	tmp = 0.0;
	if (x <= -4.5e-207)
		tmp = (2.0 + (x * (x * ((1.0 / eps_m) * (eps_m * (eps_m * eps_m)))))) / 2.0;
	elseif (x <= 7e-255)
		tmp = (2.0 + ((x * eps_m) * (x * eps_m))) / 2.0;
	elseif (x <= 335.0)
		tmp = (2.0 + t_0) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-207], N[(N[(2.0 + N[(x * N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] * N[(eps$95$m * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e-255], N[(N[(2.0 + N[(N[(x * eps$95$m), $MachinePrecision] * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 335.0], N[(N[(2.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.49999999999999992e-207

   1. Initial program 72.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 92.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 92.6%

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

   8. Simplified 92.6%

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

      1. pow2 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. inv-pow N/A

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

      5. cube-unmult N/A

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

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

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

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

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

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

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

      9. *-lowering-*.f64 96.8%

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

   10. Applied egg-rr 96.8%

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

   if -4.49999999999999992e-207 < x < 6.99999999999999958e-255

   1. Initial program 64.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 83.4%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 83.4%

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

   8. Simplified 83.4%

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

      1. associate-*r* N/A

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

      2. unswap-sqr N/A

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

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

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

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

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

      5. *-lowering-*.f64 97.2%

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

   10. Applied egg-rr 97.2%

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

   if 6.99999999999999958e-255 < x < 335

   1. Initial program 50.7%

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

   3. Taylor expanded in 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 91.5%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 91.5%

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

   8. Simplified 91.5%

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

   if 335 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 54.3%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.1%

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

   8. Simplified 54.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 55.2%

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

   11. Simplified 55.2%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right), 2\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), 2\right) \]

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

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

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

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

       6. *-lowering-*.f64 73.7%

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

   13. Applied egg-rr 73.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-207}:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(\frac{1}{\varepsilon} \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-255}:\\ \;\;\;\;\frac{2 + \left(x \cdot \varepsilon\right) \cdot \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 335:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.45e-47)
   (/ (* (* eps_m eps_m) (* x x)) 2.0)
   (if (<= x 7.4e-30) 1.0 (/ (* x (* x (* eps_m eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.45e-47) {
		tmp = ((eps_m * eps_m) * (x * x)) / 2.0;
	} else if (x <= 7.4e-30) {
		tmp = 1.0;
	} else {
		tmp = (x * (x * (eps_m * 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 (x <= (-1.45d-47)) then
        tmp = ((eps_m * eps_m) * (x * x)) / 2.0d0
    else if (x <= 7.4d-30) then
        tmp = 1.0d0
    else
        tmp = (x * (x * (eps_m * 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 (x <= -1.45e-47) {
		tmp = ((eps_m * eps_m) * (x * x)) / 2.0;
	} else if (x <= 7.4e-30) {
		tmp = 1.0;
	} else {
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.45e-47:
		tmp = ((eps_m * eps_m) * (x * x)) / 2.0
	elif x <= 7.4e-30:
		tmp = 1.0
	else:
		tmp = (x * (x * (eps_m * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.45e-47)
		tmp = Float64(Float64(Float64(eps_m * eps_m) * Float64(x * x)) / 2.0);
	elseif (x <= 7.4e-30)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * Float64(x * Float64(eps_m * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.45e-47)
		tmp = ((eps_m * eps_m) * (x * x)) / 2.0;
	elseif (x <= 7.4e-30)
		tmp = 1.0;
	else
		tmp = (x * (x * (eps_m * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.45e-47], N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.4e-30], 1.0, N[(N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.45e-47

   1. Initial program 91.9%

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

   3. Taylor expanded in 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 90.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 74.6%

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

   11. Simplified 74.6%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 2\right) \]

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

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

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

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

       5. *-lowering-*.f64 82.1%

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

   13. Applied egg-rr 82.1%

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

   if -1.45e-47 < x < 7.4000000000000006e-30

   1. Initial program 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 78.7%

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

   if 7.4000000000000006e-30 < x

   1. Initial program 97.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 55.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.7%

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

   8. Simplified 55.7%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 53.2%

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

   11. Simplified 53.2%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right), 2\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), 2\right) \]

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

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

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

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

       6. *-lowering-*.f64 71.2%

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

   13. Applied egg-rr 71.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-47}:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-30}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\left(eps\_m \cdot eps\_m\right) \cdot \left(x \cdot x\right)}{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* (* eps_m eps_m) (* x x)) 2.0)))
   (if (<= x -2.6e-45) t_0 (if (<= x 1.05e-29) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((eps_m * eps_m) * (x * x)) / 2.0;
	double tmp;
	if (x <= -2.6e-45) {
		tmp = t_0;
	} else if (x <= 1.05e-29) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((eps_m * eps_m) * (x * x)) / 2.0d0
    if (x <= (-2.6d-45)) then
        tmp = t_0
    else if (x <= 1.05d-29) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((eps_m * eps_m) * (x * x)) / 2.0;
	double tmp;
	if (x <= -2.6e-45) {
		tmp = t_0;
	} else if (x <= 1.05e-29) {
		tmp = 1.0;
	} 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)) / 2.0
	tmp = 0
	if x <= -2.6e-45:
		tmp = t_0
	elif x <= 1.05e-29:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(eps_m * eps_m) * Float64(x * x)) / 2.0)
	tmp = 0.0
	if (x <= -2.6e-45)
		tmp = t_0;
	elseif (x <= 1.05e-29)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((eps_m * eps_m) * (x * x)) / 2.0;
	tmp = 0.0;
	if (x <= -2.6e-45)
		tmp = t_0;
	elseif (x <= 1.05e-29)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.6e-45], t$95$0, If[LessEqual[x, 1.05e-29], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999987e-45 or 1.04999999999999995e-29 < x

   1. Initial program 95.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 68.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 68.5%

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

   8. Simplified 68.5%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 61.2%

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

   11. Simplified 61.2%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 2\right) \]

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

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

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

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

       5. *-lowering-*.f64 66.7%

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

   13. Applied egg-rr 66.7%

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

   if -2.59999999999999987e-45 < x < 1.04999999999999995e-29

   1. Initial program 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 78.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{eps\_m \cdot \left(eps\_m \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot eps\_m\right) \cdot \left(x \cdot eps\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7.5e-35)
   (/ (* eps_m (* eps_m (* x x))) 2.0)
   (if (<= x 3.7e-28) 1.0 (/ (* (* x eps_m) (* x eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7.5e-35) {
		tmp = (eps_m * (eps_m * (x * x))) / 2.0;
	} else if (x <= 3.7e-28) {
		tmp = 1.0;
	} else {
		tmp = ((x * eps_m) * (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 (x <= (-7.5d-35)) then
        tmp = (eps_m * (eps_m * (x * x))) / 2.0d0
    else if (x <= 3.7d-28) then
        tmp = 1.0d0
    else
        tmp = ((x * eps_m) * (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 (x <= -7.5e-35) {
		tmp = (eps_m * (eps_m * (x * x))) / 2.0;
	} else if (x <= 3.7e-28) {
		tmp = 1.0;
	} else {
		tmp = ((x * eps_m) * (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -7.5e-35:
		tmp = (eps_m * (eps_m * (x * x))) / 2.0
	elif x <= 3.7e-28:
		tmp = 1.0
	else:
		tmp = ((x * eps_m) * (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7.5e-35)
		tmp = Float64(Float64(eps_m * Float64(eps_m * Float64(x * x))) / 2.0);
	elseif (x <= 3.7e-28)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x * eps_m) * 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 (x <= -7.5e-35)
		tmp = (eps_m * (eps_m * (x * x))) / 2.0;
	elseif (x <= 3.7e-28)
		tmp = 1.0;
	else
		tmp = ((x * eps_m) * (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -7.5e-35], N[(N[(eps$95$m * N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.7e-28], 1.0, N[(N[(N[(x * eps$95$m), $MachinePrecision] * N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e-35

   1. Initial program 97.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 90.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 90.8%

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

   8. Simplified 90.8%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 84.1%

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

   11. Simplified 84.1%

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

   if -7.5e-35 < x < 3.7000000000000002e-28

   1. Initial program 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 76.9%

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

   if 3.7000000000000002e-28 < x

   1. Initial program 97.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 55.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 55.7%

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

   8. Simplified 55.7%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 53.2%

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

   11. Simplified 53.2%

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

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

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

       5. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), 2\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       7. swap-sqr N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot x\right) \cdot \left(\varepsilon \cdot x\right)\right), 2\right) \]

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

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

       9. *-commutative N/A

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

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

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

       11. *-commutative N/A

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

       12. *-lowering-*.f64 53.5%

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

   13. Applied egg-rr 53.5%

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

Alternative 9: 68.4% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{eps\_m \cdot \left(eps\_m \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-28}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* eps_m (* eps_m (* x x))) 2.0)))
   (if (<= x -8.5e-35) t_0 (if (<= x 5e-28) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (eps_m * (eps_m * (x * x))) / 2.0;
	double tmp;
	if (x <= -8.5e-35) {
		tmp = t_0;
	} else if (x <= 5e-28) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (eps_m * (eps_m * (x * x))) / 2.0d0
    if (x <= (-8.5d-35)) then
        tmp = t_0
    else if (x <= 5d-28) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (eps_m * (eps_m * (x * x))) / 2.0;
	double tmp;
	if (x <= -8.5e-35) {
		tmp = t_0;
	} else if (x <= 5e-28) {
		tmp = 1.0;
	} 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))) / 2.0
	tmp = 0
	if x <= -8.5e-35:
		tmp = t_0
	elif x <= 5e-28:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(eps_m * Float64(eps_m * Float64(x * x))) / 2.0)
	tmp = 0.0
	if (x <= -8.5e-35)
		tmp = t_0;
	elseif (x <= 5e-28)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (eps_m * (eps_m * (x * x))) / 2.0;
	tmp = 0.0;
	if (x <= -8.5e-35)
		tmp = t_0;
	elseif (x <= 5e-28)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(eps$95$m * N[(eps$95$m * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -8.5e-35], t$95$0, If[LessEqual[x, 5e-28], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000001e-35 or 5.0000000000000002e-28 < x

   1. Initial program 97.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 67.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.5%

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

   8. Simplified 67.5%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 63.6%

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

   11. Simplified 63.6%

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

   if -8.5000000000000001e-35 < x < 5.0000000000000002e-28

   1. Initial program 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 76.9%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.8% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 31000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 31000000000.0)
   1.0
   (if (<= x 2.5e+166) 0.0 (/ (+ 2.0 (* x eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 31000000000.0) {
		tmp = 1.0;
	} else if (x <= 2.5e+166) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (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 (x <= 31000000000.0d0) then
        tmp = 1.0d0
    else if (x <= 2.5d+166) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (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 (x <= 31000000000.0) {
		tmp = 1.0;
	} else if (x <= 2.5e+166) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 31000000000.0:
		tmp = 1.0
	elif x <= 2.5e+166:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 31000000000.0)
		tmp = 1.0;
	elseif (x <= 2.5e+166)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + 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 (x <= 31000000000.0)
		tmp = 1.0;
	elseif (x <= 2.5e+166)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 31000000000.0], 1.0, If[LessEqual[x, 2.5e+166], 0.0, N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1e10

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 58.7%

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

   if 3.1e10 < x < 2.5000000000000001e166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. rec-exp N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot 0}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \frac{0}{\varepsilon} \]

      8. div0 49.1%

         \[\leadsto 0 \]

   7. Simplified 49.1%

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

   if 2.5000000000000001e166 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 62.6%

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

   6. Taylor expanded in eps around 0

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

      1. /-lowering-/.f64 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in eps around inf

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

       1. +-commutative N/A

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

       2. distribute-rgt-in N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lft-mult-inverse N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. distribute-rgt-in N/A

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

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

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

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

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

       12. *-lowering-*.f64 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

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

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

       3. *-lowering-*.f64 15.3%

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

   14. Simplified 15.3%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot x + 2}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 31000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.6% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\\ \mathbf{if}\;x \leq 145:\\ \;\;\;\;\frac{2 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* x (* eps_m eps_m)))))
   (if (<= x 145.0) (/ (+ 2.0 t_0) 2.0) (/ t_0 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (x * (eps_m * eps_m));
	double tmp;
	if (x <= 145.0) {
		tmp = (2.0 + t_0) / 2.0;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (eps_m * eps_m))
    if (x <= 145.0d0) then
        tmp = (2.0d0 + t_0) / 2.0d0
    else
        tmp = t_0 / 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 = x * (x * (eps_m * eps_m));
	double tmp;
	if (x <= 145.0) {
		tmp = (2.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (x * (eps_m * eps_m))
	tmp = 0
	if x <= 145.0:
		tmp = (2.0 + t_0) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(x * Float64(eps_m * eps_m)))
	tmp = 0.0
	if (x <= 145.0)
		tmp = Float64(Float64(2.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (x * (eps_m * eps_m));
	tmp = 0.0;
	if (x <= 145.0)
		tmp = (2.0 + t_0) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 145.0], N[(N[(2.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 145

   1. Initial program 63.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 90.5%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 90.5%

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

   8. Simplified 90.5%

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

   if 145 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 54.3%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 54.1%

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

   8. Simplified 54.1%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 55.2%

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

   11. Simplified 55.2%

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

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right), 2\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot x\right), 2\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right), 2\right) \]

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

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

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

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

       6. *-lowering-*.f64 73.7%

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

   13. Applied egg-rr 73.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 145:\\ \;\;\;\;\frac{2 + x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.1% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 31000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 31000000000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 31000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 31000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 31000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 31000000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 31000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 31000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 31000000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e10

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 58.7%

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

   if 3.1e10 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. neg-mul-1 N/A

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

      3. rec-exp N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-out N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot 0}{\varepsilon} \]

      7. mul0-rgt N/A

         \[\leadsto \frac{0}{\varepsilon} \]

      8. div0 43.4%

         \[\leadsto 0 \]

   7. Simplified 43.4%

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

Alternative 13: 15.8% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 74.2%

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

3. Simplified 74.3%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. rec-exp N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-out N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{e^{\mathsf{neg}\left(x\right)} \cdot 0}{\varepsilon} \]

   7. mul0-rgt N/A

      \[\leadsto \frac{0}{\varepsilon} \]

   8. div0 14.1%

      \[\leadsto 0 \]

7. Simplified 14.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024192 
(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))