NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.8%

Time: 15.8s

Alternatives: 18

Speedup: 2.0×

• 39.1% 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: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(x, eps\_m, x\right)}\\ \mathbf{if}\;eps\_m \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{{\left({\left(e^{\sqrt[3]{{t\_0}^{4}}}\right)}^{\left({\left(\sqrt[3]{t\_0}\right)}^{2}\right)}\right)}^{t\_0}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (cbrt (fma x eps_m x))))
   (if (<= eps_m 1.05e-47)
     (/ (+ x 1.0) (exp x))
     (/
      (+
       (exp (* x (+ eps_m -1.0)))
       (/ 1.0 (pow (pow (exp (cbrt (pow t_0 4.0))) (pow (cbrt t_0) 2.0)) t_0)))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = cbrt(fma(x, eps_m, x));
	double tmp;
	if (eps_m <= 1.05e-47) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / pow(pow(exp(cbrt(pow(t_0, 4.0))), pow(cbrt(t_0), 2.0)), t_0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = cbrt(fma(x, eps_m, x))
	tmp = 0.0
	if (eps_m <= 1.05e-47)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / ((exp(cbrt((t_0 ^ 4.0))) ^ (cbrt(t_0) ^ 2.0)) ^ t_0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Power[N[(x * eps$95$m + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[eps$95$m, 1.05e-47], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[N[Power[N[Exp[N[Power[N[Power[t$95$0, 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.05e-47

   1. Initial program 69.4%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around 0 31.5%

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

      1. associate-+r+ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. sub-neg 62.0%

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

      4. +-inverses 62.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 62.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 62.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 62.6%

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

      1. +-commutative 62.6%

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

   10. Simplified 62.6%

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

       1. *-commutative 62.6%

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

       2. exp-neg 62.6%

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

       3. un-div-inv 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 1.05e-47 < eps

   1. Initial program 90.5%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. exp-prod 99.9%

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

      3. pow2 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. fma-define 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-define 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. exp-prod 99.9%

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

      3. cbrt-unprod 100.0%

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

      4. pow-prod-up 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. cbrt-prod 100.0%

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

      8. pow2 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.05 \cdot 10^{-47}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{\left({\left(e^{\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}^{4}}}\right)}^{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(x, \varepsilon, x\right)}}\right)}^{2}\right)}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left(x, eps\_m, x\right)}\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{{\left({\left(e^{t\_0}\right)}^{t\_0}\right)}^{t\_0}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (cbrt (fma x eps_m x))))
   (if (<= eps_m 2e-48)
     (/ (+ x 1.0) (exp x))
     (/
      (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (pow (pow (exp t_0) t_0) t_0)))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = cbrt(fma(x, eps_m, x));
	double tmp;
	if (eps_m <= 2e-48) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / pow(pow(exp(t_0), t_0), t_0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = cbrt(fma(x, eps_m, x))
	tmp = 0.0
	if (eps_m <= 2e-48)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / ((exp(t_0) ^ t_0) ^ t_0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Power[N[(x * eps$95$m + x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-48], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[N[Power[N[Exp[t$95$0], $MachinePrecision], t$95$0], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.9999999999999999e-48

   1. Initial program 69.4%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around 0 31.5%

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

      1. associate-+r+ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. sub-neg 62.0%

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

      4. +-inverses 62.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 62.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 62.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 62.6%

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

      1. +-commutative 62.6%

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

   10. Simplified 62.6%

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

       1. *-commutative 62.6%

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

       2. exp-neg 62.6%

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

       3. un-div-inv 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 1.9999999999999999e-48 < eps

   1. Initial program 90.5%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. exp-prod 99.9%

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

      3. pow2 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. fma-define 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-define 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. unpow2 99.9%

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

      2. exp-prod 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{\left({\left(e^{\sqrt[3]{\mathsf{fma}\left(x, \varepsilon, x\right)}}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}\right)}^{\left(\sqrt[3]{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \sqrt[3]{eps\_m \cdot x}\\ \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{{\left(e^{{t\_0}^{2}}\right)}^{t\_0}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (cbrt (* eps_m x))))
   (if (<= eps_m 5e-46)
     (/ (+ x 1.0) (exp x))
     (/
      (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (pow (exp (pow t_0 2.0)) t_0)))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = cbrt((eps_m * x));
	double tmp;
	if (eps_m <= 5e-46) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / pow(exp(pow(t_0, 2.0)), t_0))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.cbrt((eps_m * x));
	double tmp;
	if (eps_m <= 5e-46) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.pow(Math.exp(Math.pow(t_0, 2.0)), t_0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = cbrt(Float64(eps_m * x))
	tmp = 0.0
	if (eps_m <= 5e-46)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / (exp((t_0 ^ 2.0)) ^ t_0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Power[N[(eps$95$m * x), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[eps$95$m, 5e-46], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[N[Exp[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.99999999999999992e-46

   1. Initial program 69.4%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around 0 31.5%

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

      1. associate-+r+ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. sub-neg 62.0%

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

      4. +-inverses 62.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 62.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 62.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 62.6%

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

      1. +-commutative 62.6%

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

   10. Simplified 62.6%

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

       1. *-commutative 62.6%

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

       2. exp-neg 62.6%

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

       3. un-div-inv 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 4.99999999999999992e-46 < eps

   1. Initial program 90.5%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. exp-prod 99.9%

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

      3. pow2 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. fma-define 99.9%

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

      7. +-commutative 99.9%

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

      8. *-commutative 99.9%

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

      9. fma-define 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in eps around inf 99.9%

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

       1. *-commutative 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{\left(e^{{\left(\sqrt[3]{\varepsilon \cdot x}\right)}^{2}}\right)}^{\left(\sqrt[3]{\varepsilon \cdot x}\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 10^{-47}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{{\left(\sqrt{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}\right)}^{2}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1e-47)
   (/ (+ x 1.0) (exp x))
   (/
    (+
     (exp (* x (+ eps_m -1.0)))
     (/ 1.0 (pow (sqrt (exp (fma x eps_m x))) 2.0)))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-47) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / pow(sqrt(exp(fma(x, eps_m, x))), 2.0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1e-47)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / (sqrt(exp(fma(x, eps_m, x))) ^ 2.0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1e-47], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[N[Sqrt[N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.9999999999999997e-48

   1. Initial program 69.4%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around 0 31.5%

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

      1. associate-+r+ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. sub-neg 62.0%

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

      4. +-inverses 62.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 62.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 62.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 62.6%

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

      1. +-commutative 62.6%

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

   10. Simplified 62.6%

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

       1. *-commutative 62.6%

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

       2. exp-neg 62.6%

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

       3. un-div-inv 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 9.9999999999999997e-48 < eps

   1. Initial program 90.5%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. +-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-define 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-47}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{\left(\sqrt{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}\right)}^{2}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{eps\_m \cdot x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-46)
   (/ (+ x 1.0) (exp x))
   (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (* eps_m x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-46) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((eps_m * x)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5d-46) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / exp((eps_m * x)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-46) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.exp((eps_m * x)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5e-46:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp((eps_m * x)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-46)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(eps_m * x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5e-46)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((eps_m * x)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-46], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.99999999999999992e-46

   1. Initial program 69.4%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around 0 31.5%

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

      1. associate-+r+ 62.0%

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

      2. mul-1-neg 62.0%

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

      3. sub-neg 62.0%

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

      4. +-inverses 62.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 62.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 62.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 62.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 62.6%

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

      1. +-commutative 62.6%

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

   10. Simplified 62.6%

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

       1. *-commutative 62.6%

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

       2. exp-neg 62.6%

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

       3. un-div-inv 62.6%

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

   12. Applied egg-rr 62.6%

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

   if 4.99999999999999992e-46 < eps

   1. Initial program 90.5%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-46}:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{\varepsilon \cdot x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.5:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.5)
   (/ (+ x 1.0) (exp x))
   (/ (* eps_m (+ x (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.5) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5.5d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (eps_m * (x + ((1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.5) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (eps_m * (x + ((1.0 + Math.exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5.5:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (eps_m * (x + ((1.0 + math.exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5.5)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5.5)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5.5], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * N[(x + N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.5

   1. Initial program 67.2%

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

   3. Simplified 59.9%

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

   5. Taylor expanded in eps around 0 31.8%

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

      1. associate-+r+ 64.4%

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

      2. mul-1-neg 64.4%

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

      3. sub-neg 64.4%

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

      4. +-inverses 64.4%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 64.4%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 65.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 65.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 65.0%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 65.0%

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

      1. +-commutative 65.0%

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

   10. Simplified 65.0%

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

       1. *-commutative 65.0%

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

       2. exp-neg 65.0%

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

       3. un-div-inv 65.0%

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

   12. Applied egg-rr 65.0%

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

   if 5.5 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 51.1%

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

      1. associate-+r+ 51.1%

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

      2. mul-1-neg 51.1%

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

      3. associate-*r* 51.1%

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

      4. distribute-rgt-neg-in 51.1%

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

      5. remove-double-neg 51.1%

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

      6. sub-neg 51.1%

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

      7. mul-1-neg 51.1%

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

      8. associate-*r/ 51.1%

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

      9. metadata-eval 51.1%

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

      10. sub-neg 51.1%

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

      11. distribute-neg-in 51.1%

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

      12. metadata-eval 51.1%

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

      13. remove-double-neg 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in eps around inf 76.8%

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

   10. Simplified 76.8%

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

Alternative 7: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + eps\_m \cdot x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-288)
   (/ (+ 1.0 (/ 1.0 (exp (+ x (* eps_m x))))) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-288) {
		tmp = (1.0 + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-288)) then
        tmp = (1.0d0 + (1.0d0 / exp((x + (eps_m * x))))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-288) {
		tmp = (1.0 + (1.0 / Math.exp((x + (eps_m * x))))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-288:
		tmp = (1.0 + (1.0 / math.exp((x + (eps_m * x))))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-288)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x + Float64(eps_m * x))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-288)
		tmp = (1.0 + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-288], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000006e-288

   1. Initial program 69.3%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in eps around inf 96.1%

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

   6. Taylor expanded in x around 0 68.0%

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

   if -1.00000000000000006e-288 < x

   1. Initial program 80.4%

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

   3. Simplified 80.4%

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

   5. Taylor expanded in x around 0 32.3%

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

   6. Taylor expanded in eps around inf 50.8%

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

      1. associate-*r* 50.8%

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

      2. neg-mul-1 50.8%

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

      3. *-commutative 50.8%

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

   8. Simplified 50.8%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-288}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.011:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;eps\_m \leq 4.7 \cdot 10^{+198}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(eps\_m + \frac{eps\_m}{x}\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.011)
   (/ (+ x 1.0) (exp x))
   (if (<= eps_m 4.7e+198)
     (* (+ x 1.0) (exp x))
     (/ (/ (* (* x 2.0) (+ eps_m (/ eps_m x))) eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.011) {
		tmp = (x + 1.0) / exp(x);
	} else if (eps_m <= 4.7e+198) {
		tmp = (x + 1.0) * exp(x);
	} else {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.011d0) then
        tmp = (x + 1.0d0) / exp(x)
    else if (eps_m <= 4.7d+198) then
        tmp = (x + 1.0d0) * exp(x)
    else
        tmp = (((x * 2.0d0) * (eps_m + (eps_m / x))) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.011) {
		tmp = (x + 1.0) / Math.exp(x);
	} else if (eps_m <= 4.7e+198) {
		tmp = (x + 1.0) * Math.exp(x);
	} else {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.011:
		tmp = (x + 1.0) / math.exp(x)
	elif eps_m <= 4.7e+198:
		tmp = (x + 1.0) * math.exp(x)
	else:
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.011)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	elseif (eps_m <= 4.7e+198)
		tmp = Float64(Float64(x + 1.0) * exp(x));
	else
		tmp = Float64(Float64(Float64(Float64(x * 2.0) * Float64(eps_m + Float64(eps_m / x))) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.011)
		tmp = (x + 1.0) / exp(x);
	elseif (eps_m <= 4.7e+198)
		tmp = (x + 1.0) * exp(x);
	else
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.011], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 4.7e+198], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * 2.0), $MachinePrecision] * N[(eps$95$m + N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.010999999999999999

   1. Initial program 67.0%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in eps around 0 31.4%

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

      1. associate-+r+ 64.3%

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

      2. mul-1-neg 64.3%

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

      3. sub-neg 64.3%

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

      4. +-inverses 64.3%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 64.3%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 64.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 64.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 64.8%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 64.8%

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

      1. +-commutative 64.8%

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

   10. Simplified 64.8%

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

       1. *-commutative 64.8%

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

       2. exp-neg 64.8%

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

       3. un-div-inv 64.8%

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

   12. Applied egg-rr 64.8%

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

   if 0.010999999999999999 < eps < 4.7000000000000002e198

   1. Initial program 99.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in eps around 0 28.6%

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

      1. associate-+r+ 28.6%

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

      2. mul-1-neg 28.6%

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

      3. sub-neg 28.6%

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

      4. +-inverses 28.6%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 28.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 28.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 28.6%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 28.6%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 28.6%

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

      1. +-commutative 28.6%

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

   10. Simplified 28.6%

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

       1. distribute-lft-in 28.6%

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

       2. add-sqr-sqrt 9.6%

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

       3. sqrt-unprod 51.5%

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

       4. sqr-neg 51.5%

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

       5. sqrt-unprod 41.9%

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

       6. add-sqr-sqrt 93.4%

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

       7. *-rgt-identity 93.4%

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

       8. add-sqr-sqrt 51.5%

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

       9. sqrt-unprod 93.4%

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

       10. sqr-neg 93.4%

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

       11. sqrt-unprod 41.9%

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

       12. add-sqr-sqrt 52.2%

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

   12. Applied egg-rr 52.2%

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

       1. *-commutative 52.2%

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

       2. distribute-lft1-in 52.2%

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

   14. Simplified 52.2%

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

   if 4.7000000000000002e198 < 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%

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

   5. Taylor expanded in eps around 0 1.7%

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

      1. associate-+r+ 1.7%

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

      2. mul-1-neg 1.7%

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

      3. sub-neg 1.7%

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

      4. +-inverses 1.7%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 1.7%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 1.7%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 1.7%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 1.7%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in x around 0 37.0%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{1}\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in x around inf 71.6%

      \[\leadsto \frac{\frac{0 + \color{blue}{x \cdot \left(2 \cdot \varepsilon + 2 \cdot \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]
   10. Step-by-step derivation

       1. distribute-lft-out 71.6%

          \[\leadsto \frac{\frac{0 + x \cdot \color{blue}{\left(2 \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)\right)}}{\varepsilon}}{2} \]

       2. associate-*r* 71.6%

          \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   11. Simplified 71.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.011:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 4.7 \cdot 10^{+198}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-207}:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(eps\_m + \frac{eps\_m}{x}\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.45e-207)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 3.8e+22)
     (/ (/ (* (* x 2.0) (+ eps_m (/ eps_m x))) eps_m) 2.0)
     (* (+ x 1.0) (exp x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.45e-207) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 3.8e+22) {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	} else {
		tmp = (x + 1.0) * exp(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 <= 1.45d-207) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 3.8d+22) then
        tmp = (((x * 2.0d0) * (eps_m + (eps_m / x))) / eps_m) / 2.0d0
    else
        tmp = (x + 1.0d0) * exp(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 <= 1.45e-207) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 3.8e+22) {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	} else {
		tmp = (x + 1.0) * Math.exp(x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.45e-207:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 3.8e+22:
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0
	else:
		tmp = (x + 1.0) * math.exp(x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.45e-207)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 3.8e+22)
		tmp = Float64(Float64(Float64(Float64(x * 2.0) * Float64(eps_m + Float64(eps_m / x))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(x + 1.0) * exp(x));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.45e-207)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 3.8e+22)
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	else
		tmp = (x + 1.0) * exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.45e-207], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e+22], N[(N[(N[(N[(x * 2.0), $MachinePrecision] * N[(eps$95$m + N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.45000000000000006e-207

   1. Initial program 64.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 49.6%

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

   6. Taylor expanded in eps around 0 63.5%

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

   7. Taylor expanded in eps around inf 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   9. Simplified 63.5%

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

   if 1.45000000000000006e-207 < x < 3.8000000000000004e22

   1. Initial program 63.4%

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

   3. Simplified 46.8%

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

   5. Taylor expanded in eps around 0 23.2%

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

      1. associate-+r+ 59.7%

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

      2. mul-1-neg 59.7%

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

      3. sub-neg 59.7%

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

      4. +-inverses 59.7%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 59.7%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 59.7%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 59.7%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 59.7%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in x around 0 53.9%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{1}\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in x around inf 70.3%

      \[\leadsto \frac{\frac{0 + \color{blue}{x \cdot \left(2 \cdot \varepsilon + 2 \cdot \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]
   10. Step-by-step derivation

       1. distribute-lft-out 70.3%

          \[\leadsto \frac{\frac{0 + x \cdot \color{blue}{\left(2 \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)\right)}}{\varepsilon}}{2} \]

       2. associate-*r* 70.3%

          \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   11. Simplified 70.3%

       \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   if 3.8000000000000004e22 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.3%

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

      1. associate-+r+ 47.3%

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

      2. mul-1-neg 47.3%

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

      3. sub-neg 47.3%

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

      4. +-inverses 47.3%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 47.3%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 47.3%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 47.3%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 47.3%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 47.3%

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

      1. +-commutative 47.3%

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

   10. Simplified 47.3%

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

       1. distribute-lft-in 47.3%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 54.3%

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

       4. sqr-neg 54.3%

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

       5. sqrt-unprod 54.3%

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

       6. add-sqr-sqrt 54.3%

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

       7. *-rgt-identity 54.3%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 54.3%

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

       10. sqr-neg 54.3%

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

       11. sqrt-unprod 54.3%

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

       12. add-sqr-sqrt 54.3%

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

   12. Applied egg-rr 54.3%

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

       1. *-commutative 54.3%

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

       2. distribute-lft1-in 54.3%

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

   14. Simplified 54.3%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-207}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.011:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.011)
   (/ (+ x 1.0) (exp x))
   (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.011) {
		tmp = (x + 1.0) / exp(x);
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.011d0) then
        tmp = (x + 1.0d0) / exp(x)
    else
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.011) {
		tmp = (x + 1.0) / Math.exp(x);
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.011:
		tmp = (x + 1.0) / math.exp(x)
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.011)
		tmp = Float64(Float64(x + 1.0) / exp(x));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.011)
		tmp = (x + 1.0) / exp(x);
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.011], N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.010999999999999999

   1. Initial program 67.0%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in eps around 0 31.4%

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

      1. associate-+r+ 64.3%

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

      2. mul-1-neg 64.3%

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

      3. sub-neg 64.3%

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

      4. +-inverses 64.3%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 64.3%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 64.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 64.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 64.8%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 64.8%

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

      1. +-commutative 64.8%

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

   10. Simplified 64.8%

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

       1. *-commutative 64.8%

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

       2. exp-neg 64.8%

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

       3. un-div-inv 64.8%

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

   12. Applied egg-rr 64.8%

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

   if 0.010999999999999999 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 61.8%

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

   6. Taylor expanded in eps around inf 61.8%

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

      1. associate-*r* 61.8%

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

      2. neg-mul-1 61.8%

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

      3. *-commutative 61.8%

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

   8. Simplified 61.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.011:\\ \;\;\;\;\frac{x + 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-207}:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 465:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(eps\_m + \frac{eps\_m}{x}\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+192}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 - x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 7e-207)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 465.0)
     (/ (/ (* (* x 2.0) (+ eps_m (/ eps_m x))) eps_m) 2.0)
     (if (<= x 5e+192)
       0.0
       (/
        (/
         (+
          (- 1.0 x)
          (- -1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))))
         eps_m)
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 7e-207) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 465.0) {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	} else if (x <= 5e+192) {
		tmp = 0.0;
	} else {
		tmp = (((1.0 - x) + (-1.0 - (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 7d-207) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 465.0d0) then
        tmp = (((x * 2.0d0) * (eps_m + (eps_m / x))) / eps_m) / 2.0d0
    else if (x <= 5d+192) then
        tmp = 0.0d0
    else
        tmp = (((1.0d0 - x) + ((-1.0d0) - (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0)))))))) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 7e-207) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 465.0) {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	} else if (x <= 5e+192) {
		tmp = 0.0;
	} else {
		tmp = (((1.0 - x) + (-1.0 - (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 7e-207:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 465.0:
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0
	elif x <= 5e+192:
		tmp = 0.0
	else:
		tmp = (((1.0 - x) + (-1.0 - (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 7e-207)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 465.0)
		tmp = Float64(Float64(Float64(Float64(x * 2.0) * Float64(eps_m + Float64(eps_m / x))) / eps_m) / 2.0);
	elseif (x <= 5e+192)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(-1.0 - Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))))) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 7e-207)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 465.0)
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	elseif (x <= 5e+192)
		tmp = 0.0;
	else
		tmp = (((1.0 - x) + (-1.0 - (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 7e-207], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 465.0], N[(N[(N[(N[(x * 2.0), $MachinePrecision] * N[(eps$95$m + N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+192], 0.0, N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 - N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.0000000000000003e-207

   1. Initial program 64.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 49.6%

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

   6. Taylor expanded in eps around 0 63.5%

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

   7. Taylor expanded in eps around inf 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   9. Simplified 63.5%

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

   if 7.0000000000000003e-207 < x < 465

   1. Initial program 62.7%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in eps around 0 21.8%

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

      1. associate-+r+ 59.0%

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

      2. mul-1-neg 59.0%

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

      3. sub-neg 59.0%

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

      4. +-inverses 59.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 59.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 58.9%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 58.9%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 58.9%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in x around 0 54.9%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{1}\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in x around inf 71.6%

      \[\leadsto \frac{\frac{0 + \color{blue}{x \cdot \left(2 \cdot \varepsilon + 2 \cdot \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]
   10. Step-by-step derivation

       1. distribute-lft-out 71.6%

          \[\leadsto \frac{\frac{0 + x \cdot \color{blue}{\left(2 \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)\right)}}{\varepsilon}}{2} \]

       2. associate-*r* 71.6%

          \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   11. Simplified 71.6%

       \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   if 465 < x < 5.00000000000000033e192

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. mul-1-neg 54.9%

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

      3. rec-exp 54.9%

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

      4. sub-neg 54.9%

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

      5. div-sub 54.9%

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

      6. mul-1-neg 54.9%

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

      7. rec-exp 54.9%

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

      8. +-inverses 54.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 5.00000000000000033e192 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 38.4%

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

      1. associate-+r+ 38.4%

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

      2. mul-1-neg 38.4%

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

      3. associate-*r* 38.4%

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

      4. distribute-rgt-neg-in 38.4%

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

      5. remove-double-neg 38.4%

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

      6. sub-neg 38.4%

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

      7. mul-1-neg 38.4%

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

      8. associate-*r/ 38.4%

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

      9. metadata-eval 38.4%

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

      10. sub-neg 38.4%

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

      11. distribute-neg-in 38.4%

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

      12. metadata-eval 38.4%

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

      13. remove-double-neg 38.4%

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

   7. Simplified 38.4%

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

   8. Taylor expanded in eps around 0 1.8%

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

      1. neg-mul-1 1.8%

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

      2. unsub-neg 1.8%

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

      3. neg-mul-1 1.8%

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

   10. Simplified 1.8%

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

   11. Taylor expanded in x around 0 35.7%

       \[\leadsto \frac{\frac{\left(1 - x\right) - \color{blue}{\left(1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)\right)}}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-207}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 465:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+192}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 - x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(eps\_m + \frac{eps\_m}{x}\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 5.6e-205)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 600.0)
     (/ (/ (* (* x 2.0) (+ eps_m (/ eps_m x))) eps_m) 2.0)
     (if (<= x 1.85e+193)
       0.0
       (* (+ x 1.0) (+ 1.0 (* x (+ -1.0 (* x 0.5)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 5.6e-205) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 600.0) {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	} else if (x <= 1.85e+193) {
		tmp = 0.0;
	} else {
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 5.6d-205) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 600.0d0) then
        tmp = (((x * 2.0d0) * (eps_m + (eps_m / x))) / eps_m) / 2.0d0
    else if (x <= 1.85d+193) then
        tmp = 0.0d0
    else
        tmp = (x + 1.0d0) * (1.0d0 + (x * ((-1.0d0) + (x * 0.5d0))))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 5.6e-205) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 600.0) {
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	} else if (x <= 1.85e+193) {
		tmp = 0.0;
	} else {
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 5.6e-205:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 600.0:
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0
	elif x <= 1.85e+193:
		tmp = 0.0
	else:
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 5.6e-205)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 600.0)
		tmp = Float64(Float64(Float64(Float64(x * 2.0) * Float64(eps_m + Float64(eps_m / x))) / eps_m) / 2.0);
	elseif (x <= 1.85e+193)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x + 1.0) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 5.6e-205)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 600.0)
		tmp = (((x * 2.0) * (eps_m + (eps_m / x))) / eps_m) / 2.0;
	elseif (x <= 1.85e+193)
		tmp = 0.0;
	else
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 5.6e-205], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 600.0], N[(N[(N[(N[(x * 2.0), $MachinePrecision] * N[(eps$95$m + N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.85e+193], 0.0, N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.59999999999999983e-205

   1. Initial program 64.9%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in x around 0 49.6%

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

   6. Taylor expanded in eps around 0 63.5%

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

   7. Taylor expanded in eps around inf 63.5%

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

      1. associate-*r* 63.5%

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

      2. neg-mul-1 63.5%

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

   9. Simplified 63.5%

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

   if 5.59999999999999983e-205 < x < 600

   1. Initial program 62.7%

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

   3. Simplified 45.8%

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

   5. Taylor expanded in eps around 0 21.8%

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

      1. associate-+r+ 59.0%

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

      2. mul-1-neg 59.0%

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

      3. sub-neg 59.0%

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

      4. +-inverses 59.0%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 59.0%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 58.9%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 58.9%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 58.9%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in x around 0 54.9%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{1}\right)\right)}{\varepsilon}}{2} \]

   9. Taylor expanded in x around inf 71.6%

      \[\leadsto \frac{\frac{0 + \color{blue}{x \cdot \left(2 \cdot \varepsilon + 2 \cdot \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]
   10. Step-by-step derivation

       1. distribute-lft-out 71.6%

          \[\leadsto \frac{\frac{0 + x \cdot \color{blue}{\left(2 \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)\right)}}{\varepsilon}}{2} \]

       2. associate-*r* 71.6%

          \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   11. Simplified 71.6%

       \[\leadsto \frac{\frac{0 + \color{blue}{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}}{\varepsilon}}{2} \]

   if 600 < x < 1.8500000000000001e193

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. mul-1-neg 54.9%

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

      3. rec-exp 54.9%

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

      4. sub-neg 54.9%

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

      5. div-sub 54.9%

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

      6. mul-1-neg 54.9%

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

      7. rec-exp 54.9%

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

      8. +-inverses 54.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.8500000000000001e193 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.8%

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

      1. associate-+r+ 38.8%

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

      2. mul-1-neg 38.8%

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

      3. sub-neg 38.8%

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

      4. +-inverses 38.8%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 38.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 38.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 38.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 38.8%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 38.8%

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

      1. +-commutative 38.8%

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

   10. Simplified 38.8%

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

   11. Taylor expanded in x around 0 62.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-205}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;\frac{\frac{\left(x \cdot 2\right) \cdot \left(\varepsilon + \frac{\varepsilon}{x}\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 210.0)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 1e+192) 0.0 (* (+ x 1.0) (+ 1.0 (* x (+ -1.0 (* x 0.5))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1e+192) {
		tmp = 0.0;
	} else {
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))));
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 210.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 1d+192) then
        tmp = 0.0d0
    else
        tmp = (x + 1.0d0) * (1.0d0 + (x * ((-1.0d0) + (x * 0.5d0))))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1e+192) {
		tmp = 0.0;
	} else {
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 210.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 1e+192:
		tmp = 0.0
	else:
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 210.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 1e+192)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x + 1.0) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 210.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 1e+192)
		tmp = 0.0;
	else
		tmp = (x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 210.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+192], 0.0, N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 210

   1. Initial program 64.3%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around 0 51.5%

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

   6. Taylor expanded in eps around 0 61.1%

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

   7. Taylor expanded in eps around inf 61.1%

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

      1. associate-*r* 61.1%

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

      2. neg-mul-1 61.1%

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

   9. Simplified 61.1%

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

   if 210 < x < 1.00000000000000004e192

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. mul-1-neg 54.9%

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

      3. rec-exp 54.9%

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

      4. sub-neg 54.9%

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

      5. div-sub 54.9%

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

      6. mul-1-neg 54.9%

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

      7. rec-exp 54.9%

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

      8. +-inverses 54.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.00000000000000004e192 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.8%

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

      1. associate-+r+ 38.8%

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

      2. mul-1-neg 38.8%

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

      3. sub-neg 38.8%

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

      4. +-inverses 38.8%

         \[\leadsto \frac{\frac{\color{blue}{0} + \varepsilon \cdot \left(2 \cdot e^{-1 \cdot x} + 2 \cdot \left(x \cdot e^{-1 \cdot x}\right)\right)}{\varepsilon}}{2} \]

      5. distribute-lft-out 38.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \color{blue}{\left(2 \cdot \left(e^{-1 \cdot x} + x \cdot e^{-1 \cdot x}\right)\right)}}{\varepsilon}}{2} \]

      6. distribute-rgt1-in 38.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \color{blue}{\left(\left(x + 1\right) \cdot e^{-1 \cdot x}\right)}\right)}{\varepsilon}}{2} \]

      7. mul-1-neg 38.8%

         \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)\right)}{\varepsilon}}{2} \]

   7. Simplified 38.8%

      \[\leadsto \frac{\color{blue}{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot e^{-x}\right)\right)}{\varepsilon}}}{2} \]

   8. Taylor expanded in eps around 0 38.8%

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

      1. +-commutative 38.8%

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

   10. Simplified 38.8%

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

   11. Taylor expanded in x around 0 62.8%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+192}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 180:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 180.0)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 1.7e+193) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 180.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1.7e+193) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 180.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 1.7d+193) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 180.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1.7e+193) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 180.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 1.7e+193:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 180.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 1.7e+193)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 180.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 1.7e+193)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 180.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.7e+193], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 180

   1. Initial program 64.3%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around 0 51.5%

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

   6. Taylor expanded in eps around 0 61.1%

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

   7. Taylor expanded in eps around inf 61.1%

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

      1. associate-*r* 61.1%

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

      2. neg-mul-1 61.1%

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

   9. Simplified 61.1%

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

   if 180 < x < 1.69999999999999993e193

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. mul-1-neg 54.9%

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

      3. rec-exp 54.9%

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

      4. sub-neg 54.9%

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

      5. div-sub 54.9%

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

      6. mul-1-neg 54.9%

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

      7. rec-exp 54.9%

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

      8. +-inverses 54.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.69999999999999993e193 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.5%

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

   7. Taylor expanded in eps around inf 36.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 180:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 2.0 x) 2.0)
   (if (<= x 5.5e+193) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 5.5e+193) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 - x) / 2.0d0
    else if (x <= 5.5d+193) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else if (x <= 5.5e+193) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	elif x <= 5.5e+193:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	elseif (x <= 5.5e+193)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	elseif (x <= 5.5e+193)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.5e+193], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 64.3%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around 0 51.5%

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

   6. Taylor expanded in eps around 0 61.1%

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

   7. Taylor expanded in eps around 0 51.8%

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

      1. neg-mul-1 51.8%

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

   9. Simplified 51.8%

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

   if 2 < x < 5.5000000000000003e193

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. mul-1-neg 54.9%

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

      3. rec-exp 54.9%

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

      4. sub-neg 54.9%

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

      5. div-sub 54.9%

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

      6. mul-1-neg 54.9%

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

      7. rec-exp 54.9%

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

      8. +-inverses 54.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 5.5000000000000003e193 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.5%

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

   7. Taylor expanded in eps around inf 36.2%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+193}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 55.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+194}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 620.0) 1.0 (if (<= x 1.18e+194) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 620.0) {
		tmp = 1.0;
	} else if (x <= 1.18e+194) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 620.0d0) then
        tmp = 1.0d0
    else if (x <= 1.18d+194) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 620.0) {
		tmp = 1.0;
	} else if (x <= 1.18e+194) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 620.0:
		tmp = 1.0
	elif x <= 1.18e+194:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 620.0)
		tmp = 1.0;
	elseif (x <= 1.18e+194)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 620.0)
		tmp = 1.0;
	elseif (x <= 1.18e+194)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 620.0], 1.0, If[LessEqual[x, 1.18e+194], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 620

   1. Initial program 64.3%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in x around 0 51.5%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 620 < x < 1.1799999999999999e194

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 54.9%

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

      1. mul-1-neg 54.9%

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

      2. mul-1-neg 54.9%

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

      3. rec-exp 54.9%

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

      4. sub-neg 54.9%

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

      5. div-sub 54.9%

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

      6. mul-1-neg 54.9%

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

      7. rec-exp 54.9%

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

      8. +-inverses 54.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 54.9%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.1799999999999999e194 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.5%

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

   7. Taylor expanded in eps around inf 36.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 620:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+194}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.5% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 580.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 580.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 580.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 580.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 580.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 580.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 580.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 580.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 580

   1. Initial program 64.3%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in x around 0 51.5%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 580 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. mul-1-neg 47.9%

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

      3. rec-exp 47.9%

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

      4. sub-neg 47.9%

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

      5. div-sub 47.9%

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

      6. mul-1-neg 47.9%

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

      7. rec-exp 47.9%

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

      8. +-inverses 47.9%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Simplified 47.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 16.0% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

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

3. Simplified 66.8%

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

5. Taylor expanded in eps around 0 17.2%

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

   1. mul-1-neg 17.2%

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

   2. mul-1-neg 17.2%

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

   3. rec-exp 17.2%

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

   4. sub-neg 17.2%

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

   5. div-sub 17.2%

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

   6. mul-1-neg 17.2%

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

   7. rec-exp 17.2%

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

   8. +-inverses 17.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Simplified 17.5%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

8. Final simplification 17.5%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(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))