NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 100.0%

Time: 13.0s

Alternatives: 9

Speedup: 14.2×

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.9:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5 + \frac{-0.5}{eps\_m}}{\left(0.25 - \frac{0.25}{eps\_m \cdot eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)}}} - e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(-0.5 + \frac{0.5}{eps\_m}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.9)
   (/ (+ x 1.0) (exp x))
   (-
    (/
     1.0
     (/
      (+ 0.5 (/ -0.5 eps_m))
      (* (- 0.25 (/ 0.25 (* eps_m eps_m))) (exp (* x (+ eps_m -1.0))))))
    (* (exp (* x (- -1.0 eps_m))) (+ -0.5 (/ 0.5 eps_m))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.9) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (1.0 / ((0.5 + (-0.5 / eps_m)) / ((0.25 - (0.25 / (eps_m * eps_m))) * exp((x * (eps_m + -1.0)))))) - (exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.9d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (1.0d0 / ((0.5d0 + ((-0.5d0) / eps_m)) / ((0.25d0 - (0.25d0 / (eps_m * eps_m))) * exp((x * (eps_m + (-1.0d0))))))) - (exp((x * ((-1.0d0) - eps_m))) * ((-0.5d0) + (0.5d0 / eps_m)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.9) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (1.0 / ((0.5 + (-0.5 / eps_m)) / ((0.25 - (0.25 / (eps_m * eps_m))) * Math.exp((x * (eps_m + -1.0)))))) - (Math.exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.9:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (1.0 / ((0.5 + (-0.5 / eps_m)) / ((0.25 - (0.25 / (eps_m * eps_m))) * math.exp((x * (eps_m + -1.0)))))) - (math.exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.9)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(0.5 + Float64(-0.5 / eps_m)) / Float64(Float64(0.25 - Float64(0.25 / Float64(eps_m * eps_m))) * exp(Float64(x * Float64(eps_m + -1.0)))))) - Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(-0.5 + Float64(0.5 / eps_m))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.9)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (1.0 / ((0.5 + (-0.5 / eps_m)) / ((0.25 - (0.25 / (eps_m * eps_m))) * exp((x * (eps_m + -1.0)))))) - (exp((x * (-1.0 - eps_m))) * (-0.5 + (0.5 / eps_m)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.9], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(0.5 + N[(-0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(0.25 - N[(0.25 / N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 + N[(0.5 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.900000000000000022

   1. Initial program 56.4%

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

   3. Simplified 56.4%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 76.8%

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

   7. Simplified 76.8%

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot e^{0 - x}} \]
   8. Step-by-step derivation

      1. exp-diff N/A

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

      2. 1-exp N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. exp-lowering-exp.f64 76.8%

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

   9. Applied egg-rr 76.8%

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

   if 0.900000000000000022 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{0.5 + \frac{-0.5}{\varepsilon}}{\left(0.25 - \frac{0.25}{\varepsilon \cdot \varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)}}}} - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{0.5}{\varepsilon} + -0.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.9:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.5 + \frac{-0.5}{\varepsilon}}{\left(0.25 - \frac{0.25}{\varepsilon \cdot \varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)}}} - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(-0.5 + \frac{0.5}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00047:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - eps\_m\right)} + e^{eps\_m \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00047)
   (/ (+ x 1.0) (exp x))
   (* 0.5 (+ (exp (* x (- -1.0 eps_m))) (exp (* eps_m x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((eps_m * x)));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.00047d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = 0.5d0 * (exp((x * ((-1.0d0) - eps_m))) + exp((eps_m * x)))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = 0.5 * (Math.exp((x * (-1.0 - eps_m))) + Math.exp((eps_m * x)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00047:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = 0.5 * (math.exp((x * (-1.0 - eps_m))) + math.exp((eps_m * x)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00047)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + exp(Float64(eps_m * x))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00047)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = 0.5 * (exp((x * (-1.0 - eps_m))) + exp((eps_m * x)));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00047], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.69999999999999986e-4

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 76.6%

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

   7. Simplified 76.6%

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot e^{0 - x}} \]
   8. Step-by-step derivation

      1. exp-diff N/A

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

      2. 1-exp N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. exp-lowering-exp.f64 76.7%

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

   9. Applied egg-rr 76.7%

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

   if 4.69999999999999986e-4 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00047:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00047:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.5\right) \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x + \frac{x + -2}{eps\_m \cdot eps\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00047)
   (/ (+ x 1.0) (exp x))
   (+
    1.0
    (* (* x 0.5) (* eps_m (* eps_m (+ x (/ (+ x -2.0) (* eps_m eps_m)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.00047d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = 1.0d0 + ((x * 0.5d0) * (eps_m * (eps_m * (x + ((x + (-2.0d0)) / (eps_m * eps_m))))))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00047:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00047)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(1.0 + Float64(Float64(x * 0.5) * Float64(eps_m * Float64(eps_m * Float64(x + Float64(Float64(x + -2.0) / Float64(eps_m * eps_m)))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00047)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00047], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * N[(eps$95$m * N[(x + N[(N[(x + -2.0), $MachinePrecision] / N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.69999999999999986e-4

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 76.6%

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

   7. Simplified 76.6%

      \[\leadsto \color{blue}{\left(x + 1\right) \cdot e^{0 - x}} \]
   8. Step-by-step derivation

      1. exp-diff N/A

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

      2. 1-exp N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. exp-lowering-exp.f64 76.7%

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

   9. Applied egg-rr 76.7%

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

   if 4.69999999999999986e-4 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 90.1%

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

   11. Taylor expanded in eps around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. associate--l+ N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. div-sub N/A

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

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

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

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

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

       11. sub-neg N/A

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

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

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 84.5%

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

   13. Simplified 84.5%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00047:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.5\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \frac{x + -2}{\varepsilon \cdot \varepsilon}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00047:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.5\right) \cdot \left(eps\_m \cdot \left(eps\_m \cdot \left(x + \frac{x + -2}{eps\_m \cdot eps\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00047)
   (exp (- 0.0 x))
   (+
    1.0
    (* (* x 0.5) (* eps_m (* eps_m (+ x (/ (+ x -2.0) (* eps_m eps_m)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = exp((0.0 - x));
	} else {
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.00047d0) then
        tmp = exp((0.0d0 - x))
    else
        tmp = 1.0d0 + ((x * 0.5d0) * (eps_m * (eps_m * (x + ((x + (-2.0d0)) / (eps_m * eps_m))))))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = Math.exp((0.0 - x));
	} else {
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00047:
		tmp = math.exp((0.0 - x))
	else:
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00047)
		tmp = exp(Float64(0.0 - x));
	else
		tmp = Float64(1.0 + Float64(Float64(x * 0.5) * Float64(eps_m * Float64(eps_m * Float64(x + Float64(Float64(x + -2.0) / Float64(eps_m * eps_m)))))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00047)
		tmp = exp((0.0 - x));
	else
		tmp = 1.0 + ((x * 0.5) * (eps_m * (eps_m * (x + ((x + -2.0) / (eps_m * eps_m))))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00047], N[Exp[N[(0.0 - x), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] * N[(eps$95$m * N[(eps$95$m * N[(x + N[(N[(x + -2.0), $MachinePrecision] / N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.69999999999999986e-4

   1. Initial program 56.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in eps around 0

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

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

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

      2. neg-mul-1 N/A

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

      3. neg-sub0 N/A

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

      4. --lowering--.f64 82.5%

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

   10. Simplified 82.5%

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

   if 4.69999999999999986e-4 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 90.1%

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

   11. Taylor expanded in eps around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. associate--l+ N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. div-sub N/A

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

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

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

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

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

       11. sub-neg N/A

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

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

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

       13. metadata-eval N/A

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

       14. unpow2 N/A

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

       15. *-lowering-*.f64 84.5%

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

   13. Simplified 84.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00047:\\ \;\;\;\;e^{0 - x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot 0.5\right) \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \left(x + \frac{x + -2}{\varepsilon \cdot \varepsilon}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.3% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot \left(x \cdot \left(eps\_m \cdot eps\_m\right)\right)\right)\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-11}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (* x (* x (* eps_m eps_m))))))
   (if (<= x -6.7e-12) t_0 (if (<= x 4.4e-11) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 0.5 * (x * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -6.7e-12) {
		tmp = t_0;
	} else if (x <= 4.4e-11) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x * (x * (eps_m * eps_m)))
    if (x <= (-6.7d-12)) then
        tmp = t_0
    else if (x <= 4.4d-11) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 0.5 * (x * (x * (eps_m * eps_m)));
	double tmp;
	if (x <= -6.7e-12) {
		tmp = t_0;
	} else if (x <= 4.4e-11) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 0.5 * (x * (x * (eps_m * eps_m)))
	tmp = 0
	if x <= -6.7e-12:
		tmp = t_0
	elif x <= 4.4e-11:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(0.5 * Float64(x * Float64(x * Float64(eps_m * eps_m))))
	tmp = 0.0
	if (x <= -6.7e-12)
		tmp = t_0;
	elseif (x <= 4.4e-11)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 0.5 * (x * (x * (eps_m * eps_m)));
	tmp = 0.0;
	if (x <= -6.7e-12)
		tmp = t_0;
	elseif (x <= 4.4e-11)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(0.5 * N[(x * N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.7e-12], t$95$0, If[LessEqual[x, 4.4e-11], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7000000000000001e-12 or 4.4000000000000003e-11 < x

   1. Initial program 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 60.9%

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

   11. Taylor expanded in eps around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 73.4%

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

   13. Simplified 73.4%

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

   if -6.7000000000000001e-12 < x < 4.4000000000000003e-11

   1. Initial program 43.7%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 84.8%

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

Alternative 6: 81.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(eps\_m \cdot eps\_m\right)\\ \mathbf{if}\;x \leq 310:\\ \;\;\;\;1 + \left(x \cdot 0.5\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (* eps_m eps_m))))
   (if (<= x 310.0) (+ 1.0 (* (* x 0.5) t_0)) (* 0.5 (* x t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (eps_m * eps_m);
	double tmp;
	if (x <= 310.0) {
		tmp = 1.0 + ((x * 0.5) * t_0);
	} else {
		tmp = 0.5 * (x * t_0);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (eps_m * eps_m)
    if (x <= 310.0d0) then
        tmp = 1.0d0 + ((x * 0.5d0) * t_0)
    else
        tmp = 0.5d0 * (x * t_0)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (eps_m * eps_m);
	double tmp;
	if (x <= 310.0) {
		tmp = 1.0 + ((x * 0.5) * t_0);
	} else {
		tmp = 0.5 * (x * t_0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (eps_m * eps_m)
	tmp = 0
	if x <= 310.0:
		tmp = 1.0 + ((x * 0.5) * t_0)
	else:
		tmp = 0.5 * (x * t_0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(eps_m * eps_m))
	tmp = 0.0
	if (x <= 310.0)
		tmp = Float64(1.0 + Float64(Float64(x * 0.5) * t_0));
	else
		tmp = Float64(0.5 * Float64(x * t_0));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (eps_m * eps_m);
	tmp = 0.0;
	if (x <= 310.0)
		tmp = 1.0 + ((x * 0.5) * t_0);
	else
		tmp = 0.5 * (x * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 310.0], N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 310

   1. Initial program 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 89.5%

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

   11. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f64 89.8%

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

   13. Simplified 89.8%

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

   if 310 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 44.9%

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

   11. Taylor expanded in eps around inf

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

       10. *-lowering-*.f64 67.3%

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

   13. Simplified 67.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 310:\\ \;\;\;\;1 + \left(x \cdot 0.5\right) \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.75) (- 1.0 x) (* x (* 0.5 (* x x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.75) {
		tmp = 1.0 - x;
	} else {
		tmp = x * (0.5 * (x * x));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.75d0) then
        tmp = 1.0d0 - x
    else
        tmp = x * (0.5d0 * (x * x))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.75) {
		tmp = 1.0 - x;
	} else {
		tmp = x * (0.5 * (x * x));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.75:
		tmp = 1.0 - x
	else:
		tmp = x * (0.5 * (x * x))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.75)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(x * Float64(0.5 * Float64(x * x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.75)
		tmp = 1.0 - x;
	else
		tmp = x * (0.5 * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.75], N[(1.0 - x), $MachinePrecision], N[(x * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 89.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + -1 \cdot x} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

       2. unsub-neg N/A

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

       3. --lowering--.f64 64.3%

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

   13. Simplified 64.3%

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

   if 0.75 < x

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. distribute-lft-out N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-rgt1-in N/A

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

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

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-sub0 N/A

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

      12. --lowering--.f64 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lowering-*.f64 36.4%

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

   10. Simplified 36.4%

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

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{3}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {x}^{3} \cdot \color{blue}{\frac{1}{2}} \]

       2. cube-mult N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

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

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 36.4%

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

   13. Simplified 36.4%

       \[\leadsto \color{blue}{x \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 50.7% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.95e-8) (- 1.0 x) (* 0.5 (* eps_m x))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.95e-8) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.95d-8) then
        tmp = 1.0d0 - x
    else
        tmp = 0.5d0 * (eps_m * x)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.95e-8) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.95e-8:
		tmp = 1.0 - x
	else:
		tmp = 0.5 * (eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.95e-8)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(0.5 * Float64(eps_m * x));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.95e-8)
		tmp = 1.0 - x;
	else
		tmp = 0.5 * (eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.95e-8], N[(1.0 - x), $MachinePrecision], N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9499999999999999e-8

   1. Initial program 55.2%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in eps around inf

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

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

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. --lowering--.f64 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in x around 0

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

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

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

      2. distribute-lft-out N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. +-commutative N/A

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

   10. Simplified 89.8%

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

   11. Taylor expanded in x around 0

       \[\leadsto \color{blue}{1 + -1 \cdot x} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

       2. unsub-neg N/A

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

       3. --lowering--.f64 65.0%

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

   13. Simplified 65.0%

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

   if 2.9499999999999999e-8 < x

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 22.1%

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

   7. Simplified 22.1%

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

   8. Taylor expanded in eps around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 13.5%

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

   10. Simplified 13.5%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.6% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 67.8%

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

3. Simplified 67.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 47.0%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))