NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 99.9%

Time: 21.0s

Alternatives: 16

Speedup: 1.9×

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

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  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: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := e^{-x}\\ t_2 := t\_1 + x \cdot t\_1\\ t_3 := e^{\mathsf{fma}\left(eps\_m, x, x\right)}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(eps\_m + -1\right)} - e^{x \cdot \left(-1 - eps\_m\right)} \cdot \left(\frac{1}{eps\_m} + -1\right) \leq 2:\\ \;\;\;\;\frac{t\_2 + t\_2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{expm1}\left(\mathsf{log1p}\left(t\_3\right)\right), \frac{1 + \frac{-1}{eps\_m}}{t\_3}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m)))
        (t_1 (exp (- x)))
        (t_2 (+ t_1 (* x t_1)))
        (t_3 (exp (fma eps_m x x))))
   (if (<=
        (-
         (* t_0 (exp (* x (+ eps_m -1.0))))
         (* (exp (* x (- -1.0 eps_m))) (+ (/ 1.0 eps_m) -1.0)))
        2.0)
     (/ (+ t_2 t_2) 2.0)
     (/ (fma t_0 (expm1 (log1p t_3)) (/ (+ 1.0 (/ -1.0 eps_m)) t_3)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = exp(-x);
	double t_2 = t_1 + (x * t_1);
	double t_3 = exp(fma(eps_m, x, x));
	double tmp;
	if (((t_0 * exp((x * (eps_m + -1.0)))) - (exp((x * (-1.0 - eps_m))) * ((1.0 / eps_m) + -1.0))) <= 2.0) {
		tmp = (t_2 + t_2) / 2.0;
	} else {
		tmp = fma(t_0, expm1(log1p(t_3)), ((1.0 + (-1.0 / eps_m)) / t_3)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_1 + Float64(x * t_1))
	t_3 = exp(fma(eps_m, x, x))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps_m + -1.0)))) - Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(Float64(1.0 / eps_m) + -1.0))) <= 2.0)
		tmp = Float64(Float64(t_2 + t_2) / 2.0);
	else
		tmp = Float64(fma(t_0, expm1(log1p(t_3)), Float64(Float64(1.0 + Float64(-1.0 / eps_m)) / t_3)) / 2.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 2

   1. Initial program 55.5%

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

      1. fma-neg 55.3%

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

      2. /-rgt-identity 55.3%

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

      3. fma-neg 55.5%

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

      4. /-rgt-identity 55.5%

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

      5. distribute-rgt-neg-in 55.5%

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

      6. sub-neg 55.5%

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

      7. metadata-eval 55.5%

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

      8. distribute-rgt-neg-in 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in eps around 0 100.0%

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

   if 2 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 99.0%

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

   3. Simplified 74.8%

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

      1. pow-exp 99.0%

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

      2. distribute-rgt-in 99.0%

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

      3. neg-mul-1 99.0%

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

      4. add-sqr-sqrt 48.6%

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

      5. sqrt-unprod 89.9%

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

      6. sqr-neg 89.9%

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

      7. sqrt-unprod 50.4%

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

      8. add-sqr-sqrt 99.0%

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

      9. fma-udef 99.0%

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

      10. expm1-log1p-u 99.0%

          \[\leadsto \frac{\mathsf{fma}\left(1 - \frac{-1}{\varepsilon}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2} \]

   6. Applied egg-rr 99.0%

      \[\leadsto \frac{\mathsf{fma}\left(1 - \frac{-1}{\varepsilon}, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{1}{\varepsilon} + -1\right) \leq 2:\\ \;\;\;\;\frac{\left(e^{-x} + x \cdot e^{-x}\right) + \left(e^{-x} + x \cdot e^{-x}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, \mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)\right), \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-269}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \left(e^{eps\_m \cdot x} - eps\_m \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.35e-269)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 2.3e+91)
     (/ (+ 1.0 (- (exp (* eps_m x)) (* eps_m x))) 2.0)
     (/ (/ (+ (exp (- x)) (/ -1.0 (exp x))) eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.35e-269) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2.3e+91) {
		tmp = (1.0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0;
	} else {
		tmp = ((exp(-x) + (-1.0 / exp(x))) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2.35d-269)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 2.3d+91) then
        tmp = (1.0d0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0d0
    else
        tmp = ((exp(-x) + ((-1.0d0) / exp(x))) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.35e-269) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2.3e+91) {
		tmp = (1.0 + (Math.exp((eps_m * x)) - (eps_m * x))) / 2.0;
	} else {
		tmp = ((Math.exp(-x) + (-1.0 / Math.exp(x))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.35e-269:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 2.3e+91:
		tmp = (1.0 + (math.exp((eps_m * x)) - (eps_m * x))) / 2.0
	else:
		tmp = ((math.exp(-x) + (-1.0 / math.exp(x))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.35e-269)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 2.3e+91)
		tmp = Float64(Float64(1.0 + Float64(exp(Float64(eps_m * x)) - Float64(eps_m * x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(-x)) + Float64(-1.0 / exp(x))) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.35e-269)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 2.3e+91)
		tmp = (1.0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0;
	else
		tmp = ((exp(-x) + (-1.0 / exp(x))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.35e-269], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.3e+91], N[(N[(1.0 + N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] + N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3499999999999999e-269

   1. Initial program 67.7%

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

      1. fma-neg 67.8%

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

      2. /-rgt-identity 67.8%

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

      3. fma-neg 67.7%

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

      4. /-rgt-identity 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. sub-neg 67.7%

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

      7. metadata-eval 67.7%

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

      8. distribute-rgt-neg-in 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in eps around inf 96.7%

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

      1. exp-prod 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 64.8%

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

   if -2.3499999999999999e-269 < x < 2.29999999999999991e91

   1. Initial program 65.2%

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

      1. fma-neg 64.8%

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

      2. /-rgt-identity 64.8%

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

      3. fma-neg 65.2%

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

      4. /-rgt-identity 65.2%

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

      5. distribute-rgt-neg-in 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around 0 46.8%

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

   6. Taylor expanded in eps around inf 80.4%

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

      1. +-commutative 80.4%

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

      2. associate-*r* 80.4%

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

      3. sub-neg 80.4%

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

      4. neg-mul-1 80.4%

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

      5. associate-*r* 80.4%

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

      6. neg-mul-1 80.4%

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

      7. distribute-lft-neg-in 80.4%

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

      8. *-commutative 80.4%

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

      9. associate-*r* 80.4%

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

      10. neg-mul-1 80.4%

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

      11. sub-neg 80.4%

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

      12. mul-1-neg 80.4%

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

   8. Simplified 80.4%

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

   9. Taylor expanded in eps around inf 80.7%

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

       1. *-commutative 80.7%

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

   11. Simplified 80.7%

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

   if 2.29999999999999991e91 < 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 0 65.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-269}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \left(e^{\varepsilon \cdot x} - \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x} \cdot \left(1 + x\right)\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (exp (- x)) (+ 1.0 x))))
   (if (<= eps_m 2e-6)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x) * (1.0 + x);
	double tmp;
	if (eps_m <= 2e-6) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x) * (1.0d0 + x)
    if (eps_m <= 2d-6) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-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 t_0 = Math.exp(-x) * (1.0 + x);
	double tmp;
	if (eps_m <= 2e-6) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x) * (1.0 + x)
	tmp = 0
	if eps_m <= 2e-6:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(exp(Float64(-x)) * Float64(1.0 + x))
	tmp = 0.0
	if (eps_m <= 2e-6)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x) * (1.0 + x);
	tmp = 0.0;
	if (eps_m <= 2e-6)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (eps_m + -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_] := Block[{t$95$0 = N[(N[Exp[(-x)], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-6], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.99999999999999991e-6

   1. Initial program 64.7%

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

      1. fma-neg 64.5%

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

      2. /-rgt-identity 64.5%

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

      3. fma-neg 64.7%

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

      4. /-rgt-identity 64.7%

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

      5. distribute-rgt-neg-in 64.7%

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

      6. sub-neg 64.7%

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

      7. metadata-eval 64.7%

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

      8. distribute-rgt-neg-in 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in eps around 0 73.1%

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

   7. Simplified 73.6%

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

   if 1.99999999999999991e-6 < eps

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(1 + x\right) + e^{-x} \cdot \left(1 + x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\frac{e^{x \cdot \left(1 + eps\_m\right)} + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.92)
   (/ (+ (exp (* x (+ 1.0 eps_m))) (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ (exp (- x)) (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.92) {
		tmp = (exp((x * (1.0 + eps_m))) + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (exp(-x) + 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 <= 0.92d0) then
        tmp = (exp((x * (1.0d0 + eps_m))) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (exp(-x) + 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 <= 0.92) {
		tmp = (Math.exp((x * (1.0 + eps_m))) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + 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 <= 0.92:
		tmp = (math.exp((x * (1.0 + eps_m))) + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (math.exp(-x) + 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 <= 0.92)
		tmp = Float64(Float64(exp(Float64(x * Float64(1.0 + eps_m))) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + 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 <= 0.92)
		tmp = (exp((x * (1.0 + eps_m))) + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (exp(-x) + 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, 0.92], N[(N[(N[Exp[N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + 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 < 0.92000000000000004

   1. Initial program 63.2%

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

      1. fma-neg 63.0%

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

      2. /-rgt-identity 63.0%

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

      3. fma-neg 63.2%

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

      4. /-rgt-identity 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. sub-neg 63.2%

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

      7. metadata-eval 63.2%

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

      8. distribute-rgt-neg-in 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in eps around inf 97.9%

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

      1. exp-prod 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. pow-exp 97.9%

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

      2. mul-1-neg 97.9%

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

      3. distribute-lft-neg-out 97.9%

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

      4. *-commutative 97.9%

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

      5. exp-prod 80.2%

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

      6. sub-neg 80.2%

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

      7. exp-sum 80.2%

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

      8. add-sqr-sqrt 44.7%

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

      9. sqrt-unprod 72.8%

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

      10. sqr-neg 72.8%

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

      11. sqrt-unprod 28.1%

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

      12. add-sqr-sqrt 55.9%

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

      13. exp-sum 55.9%

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

      14. add-sqr-sqrt 33.7%

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

      15. sqrt-unprod 76.5%

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

      16. sqr-neg 76.5%

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

      17. sqrt-unprod 31.5%

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

      18. add-sqr-sqrt 80.7%

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

      19. exp-prod 98.3%

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

      20. *-commutative 98.3%

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

   9. Applied egg-rr 98.3%

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

   if 0.92000000000000004 < x

   1. Initial program 98.8%

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

      1. fma-neg 98.8%

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

      2. /-rgt-identity 98.8%

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

      3. fma-neg 98.8%

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

      4. /-rgt-identity 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. sub-neg 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-rgt-neg-in 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around 0 80.0%

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

      1. mul-1-neg 80.0%

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

   8. Simplified 80.0%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;\frac{e^{x \cdot \left(1 + \varepsilon\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 3.85:\\ \;\;\;\;\frac{t\_0 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 3.85)
     (/ (+ t_0 (exp (* eps_m (- x)))) 2.0)
     (/ (+ (exp (- x)) t_0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 3.85) {
		tmp = (t_0 + exp((eps_m * -x))) / 2.0;
	} else {
		tmp = (exp(-x) + t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= 3.85d0) then
        tmp = (t_0 + exp((eps_m * -x))) / 2.0d0
    else
        tmp = (exp(-x) + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 3.85) {
		tmp = (t_0 + Math.exp((eps_m * -x))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 3.85:
		tmp = (t_0 + math.exp((eps_m * -x))) / 2.0
	else:
		tmp = (math.exp(-x) + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 3.85)
		tmp = Float64(Float64(t_0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 3.85)
		tmp = (t_0 + exp((eps_m * -x))) / 2.0;
	else
		tmp = (exp(-x) + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 3.85], N[(N[(t$95$0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.85000000000000009

   1. Initial program 63.2%

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

      1. fma-neg 63.0%

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

      2. /-rgt-identity 63.0%

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

      3. fma-neg 63.2%

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

      4. /-rgt-identity 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. sub-neg 63.2%

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

      7. metadata-eval 63.2%

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

      8. distribute-rgt-neg-in 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in eps around inf 97.9%

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

   6. Taylor expanded in eps around inf 97.9%

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

      1. *-commutative 97.9%

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

   8. Simplified 97.9%

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

   if 3.85000000000000009 < x

   1. Initial program 98.8%

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

      1. fma-neg 98.8%

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

      2. /-rgt-identity 98.8%

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

      3. fma-neg 98.8%

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

      4. /-rgt-identity 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. sub-neg 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-rgt-neg-in 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around 0 80.0%

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

      1. mul-1-neg 80.0%

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

   8. Simplified 80.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.85:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-287)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ (exp (- x)) (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-287) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (exp(-x) + 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-287)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (exp(-x) + 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-287) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + 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-287:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (math.exp(-x) + 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-287)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + 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-287)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (exp(-x) + 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-287], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + 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 < -1.00000000000000002e-287

   1. Initial program 66.7%

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

      1. fma-neg 66.8%

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

      2. /-rgt-identity 66.8%

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

      3. fma-neg 66.7%

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

      4. /-rgt-identity 66.7%

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

      5. distribute-rgt-neg-in 66.7%

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

      6. sub-neg 66.7%

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

      7. metadata-eval 66.7%

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

      8. distribute-rgt-neg-in 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in eps around inf 97.0%

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

      1. exp-prod 97.0%

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

   7. Applied egg-rr 97.0%

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

   8. Taylor expanded in x around 0 67.0%

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

   if -1.00000000000000002e-287 < x

   1. Initial program 78.5%

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

      1. fma-neg 78.2%

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

      2. /-rgt-identity 78.2%

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

      3. fma-neg 78.5%

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

      4. /-rgt-identity 78.5%

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

      5. distribute-rgt-neg-in 78.5%

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

      6. sub-neg 78.5%

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

      7. metadata-eval 78.5%

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

      8. distribute-rgt-neg-in 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around 0 85.2%

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

      1. mul-1-neg 85.2%

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

   8. Simplified 85.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + eps\_m\right)\\ t_1 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -550000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \left(e^{eps\_m \cdot x} - eps\_m \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+252} \lor \neg \left(x \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{t\_1 + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{eps\_m} + x \cdot \left(t\_1 \cdot \left(eps\_m + -1\right)\right)\right)\right) + \left(1 + \left(\left(t\_0 + \frac{1}{eps\_m} \cdot t\_0\right) + \frac{-1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 eps_m))) (t_1 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -550000000.0)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 9.6e+91)
       (/ (+ 1.0 (- (exp (* eps_m x)) (* eps_m x))) 2.0)
       (if (or (<= x 6e+252) (not (<= x 8.4e+273)))
         (/ (+ t_1 (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/
          (+
           (+ 1.0 (+ (/ 1.0 eps_m) (* x (* t_1 (+ eps_m -1.0)))))
           (+ 1.0 (+ (+ t_0 (* (/ 1.0 eps_m) t_0)) (/ -1.0 eps_m))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -550000000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 9.6e+91) {
		tmp = (1.0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0;
	} else if ((x <= 6e+252) || !(x <= 8.4e+273)) {
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 + eps_m)
    t_1 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-550000000.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 9.6d+91) then
        tmp = (1.0d0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0d0
    else if ((x <= 6d+252) .or. (.not. (x <= 8.4d+273))) then
        tmp = (t_1 + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = ((1.0d0 + ((1.0d0 / eps_m) + (x * (t_1 * (eps_m + (-1.0d0)))))) + (1.0d0 + ((t_0 + ((1.0d0 / eps_m) * t_0)) + ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -550000000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 9.6e+91) {
		tmp = (1.0 + (Math.exp((eps_m * x)) - (eps_m * x))) / 2.0;
	} else if ((x <= 6e+252) || !(x <= 8.4e+273)) {
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + eps_m)
	t_1 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -550000000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 9.6e+91:
		tmp = (1.0 + (math.exp((eps_m * x)) - (eps_m * x))) / 2.0
	elif (x <= 6e+252) or not (x <= 8.4e+273):
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + eps_m))
	t_1 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -550000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 9.6e+91)
		tmp = Float64(Float64(1.0 + Float64(exp(Float64(eps_m * x)) - Float64(eps_m * x))) / 2.0);
	elseif ((x <= 6e+252) || !(x <= 8.4e+273))
		tmp = Float64(Float64(t_1 + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(x * Float64(t_1 * Float64(eps_m + -1.0))))) + Float64(1.0 + Float64(Float64(t_0 + Float64(Float64(1.0 / eps_m) * t_0)) + Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (1.0 + eps_m);
	t_1 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -550000000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 9.6e+91)
		tmp = (1.0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0;
	elseif ((x <= 6e+252) || ~((x <= 8.4e+273)))
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -550000000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.6e+91], N[(N[(1.0 + N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6e+252], N[Not[LessEqual[x, 8.4e+273]], $MachinePrecision]], N[(N[(t$95$1 + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * N[(t$95$1 * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(t$95$0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.5e8

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 61.9%

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

   6. Taylor expanded in eps around inf 61.9%

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

      1. +-commutative 61.9%

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

      2. associate-*r* 61.9%

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

      3. sub-neg 61.9%

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

      4. neg-mul-1 61.9%

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

      5. associate-*r* 61.9%

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

      6. neg-mul-1 61.9%

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

      7. distribute-lft-neg-in 61.9%

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

      8. *-commutative 61.9%

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

      9. associate-*r* 61.9%

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

      10. neg-mul-1 61.9%

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

      11. sub-neg 61.9%

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

      12. mul-1-neg 61.9%

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

   8. Simplified 61.9%

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

   9. Taylor expanded in eps around 0 100.0%

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

       1. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

   if -5.5e8 < x < 9.59999999999999932e91

   1. Initial program 59.5%

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

      1. fma-neg 59.3%

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

      2. /-rgt-identity 59.3%

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

      3. fma-neg 59.5%

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

      4. /-rgt-identity 59.5%

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

      5. distribute-rgt-neg-in 59.5%

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

      6. sub-neg 59.5%

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

      7. metadata-eval 59.5%

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

      8. distribute-rgt-neg-in 59.5%

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

   3. Simplified 59.5%

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

   5. Taylor expanded in x around 0 43.7%

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

   6. Taylor expanded in eps around inf 81.0%

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

      1. +-commutative 81.0%

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

      2. associate-*r* 81.0%

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

      3. sub-neg 81.0%

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

      4. neg-mul-1 81.0%

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

      5. associate-*r* 81.0%

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

      6. neg-mul-1 81.0%

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

      7. distribute-lft-neg-in 81.0%

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

      8. *-commutative 81.0%

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

      9. associate-*r* 81.0%

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

      10. neg-mul-1 81.0%

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

      11. sub-neg 81.0%

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

      12. mul-1-neg 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in eps around inf 81.5%

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

       1. *-commutative 81.5%

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

   11. Simplified 81.5%

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

   if 9.59999999999999932e91 < x < 5.99999999999999978e252 or 8.40000000000000007e273 < x

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 17.4%

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

   6. Taylor expanded in x around 0 68.3%

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

   if 5.99999999999999978e252 < x < 8.40000000000000007e273

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 14.8%

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

   6. Taylor expanded in x around 0 28.6%

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

      1. associate-*r* 28.6%

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

      2. sub-neg 28.6%

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

      3. metadata-eval 28.6%

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

      4. distribute-rgt-in 14.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 57.6%

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

      7. mul-1-neg 57.6%

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

      8. mul-1-neg 57.6%

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

      9. sqr-neg 57.6%

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

      10. sqrt-unprod 71.7%

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

      11. add-sqr-sqrt 71.7%

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

   8. Applied egg-rr 71.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \left(e^{\varepsilon \cdot x} - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+252} \lor \neg \left(x \leq 8.4 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)\right)\right) + \left(1 + \left(\left(x \cdot \left(1 + \varepsilon\right) + \frac{1}{\varepsilon} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := x \cdot \left(1 + eps\_m\right)\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{-269}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \left(e^{eps\_m \cdot x} - eps\_m \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+252} \lor \neg \left(x \leq 2.4 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{t\_0 + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{eps\_m} + x \cdot \left(t\_0 \cdot \left(eps\_m + -1\right)\right)\right)\right) + \left(1 + \left(\left(t\_1 + \frac{1}{eps\_m} \cdot t\_1\right) + \frac{-1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))) (t_1 (* x (+ 1.0 eps_m))))
   (if (<= x -2.35e-269)
     (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
     (if (<= x 1.25e+91)
       (/ (+ 1.0 (- (exp (* eps_m x)) (* eps_m x))) 2.0)
       (if (or (<= x 8e+252) (not (<= x 2.4e+273)))
         (/ (+ t_0 (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/
          (+
           (+ 1.0 (+ (/ 1.0 eps_m) (* x (* t_0 (+ eps_m -1.0)))))
           (+ 1.0 (+ (+ t_1 (* (/ 1.0 eps_m) t_1)) (/ -1.0 eps_m))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = x * (1.0 + eps_m);
	double tmp;
	if (x <= -2.35e-269) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.25e+91) {
		tmp = (1.0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0;
	} else if ((x <= 8e+252) || !(x <= 2.4e+273)) {
		tmp = (t_0 + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_0 * (eps_m + -1.0))))) + (1.0 + ((t_1 + ((1.0 / eps_m) * t_1)) + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (1.0d0 / eps_m)
    t_1 = x * (1.0d0 + eps_m)
    if (x <= (-2.35d-269)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 1.25d+91) then
        tmp = (1.0d0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0d0
    else if ((x <= 8d+252) .or. (.not. (x <= 2.4d+273))) then
        tmp = (t_0 + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = ((1.0d0 + ((1.0d0 / eps_m) + (x * (t_0 * (eps_m + (-1.0d0)))))) + (1.0d0 + ((t_1 + ((1.0d0 / eps_m) * t_1)) + ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = x * (1.0 + eps_m);
	double tmp;
	if (x <= -2.35e-269) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.25e+91) {
		tmp = (1.0 + (Math.exp((eps_m * x)) - (eps_m * x))) / 2.0;
	} else if ((x <= 8e+252) || !(x <= 2.4e+273)) {
		tmp = (t_0 + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_0 * (eps_m + -1.0))))) + (1.0 + ((t_1 + ((1.0 / eps_m) * t_1)) + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (1.0 / eps_m)
	t_1 = x * (1.0 + eps_m)
	tmp = 0
	if x <= -2.35e-269:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 1.25e+91:
		tmp = (1.0 + (math.exp((eps_m * x)) - (eps_m * x))) / 2.0
	elif (x <= 8e+252) or not (x <= 2.4e+273):
		tmp = (t_0 + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_0 * (eps_m + -1.0))))) + (1.0 + ((t_1 + ((1.0 / eps_m) * t_1)) + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(x * Float64(1.0 + eps_m))
	tmp = 0.0
	if (x <= -2.35e-269)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.25e+91)
		tmp = Float64(Float64(1.0 + Float64(exp(Float64(eps_m * x)) - Float64(eps_m * x))) / 2.0);
	elseif ((x <= 8e+252) || !(x <= 2.4e+273))
		tmp = Float64(Float64(t_0 + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(x * Float64(t_0 * Float64(eps_m + -1.0))))) + Float64(1.0 + Float64(Float64(t_1 + Float64(Float64(1.0 / eps_m) * t_1)) + Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (1.0 / eps_m);
	t_1 = x * (1.0 + eps_m);
	tmp = 0.0;
	if (x <= -2.35e-269)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.25e+91)
		tmp = (1.0 + (exp((eps_m * x)) - (eps_m * x))) / 2.0;
	elseif ((x <= 8e+252) || ~((x <= 2.4e+273)))
		tmp = (t_0 + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_0 * (eps_m + -1.0))))) + (1.0 + ((t_1 + ((1.0 / eps_m) * t_1)) + (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.35e-269], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.25e+91], N[(N[(1.0 + N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 8e+252], N[Not[LessEqual[x, 2.4e+273]], $MachinePrecision]], N[(N[(t$95$0 + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * N[(t$95$0 * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(t$95$1 + N[(N[(1.0 / eps$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.3499999999999999e-269

   1. Initial program 67.7%

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

      1. fma-neg 67.8%

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

      2. /-rgt-identity 67.8%

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

      3. fma-neg 67.7%

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

      4. /-rgt-identity 67.7%

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

      5. distribute-rgt-neg-in 67.7%

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

      6. sub-neg 67.7%

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

      7. metadata-eval 67.7%

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

      8. distribute-rgt-neg-in 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in eps around inf 96.7%

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

      1. exp-prod 96.7%

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

   7. Applied egg-rr 96.7%

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

   8. Taylor expanded in x around 0 64.8%

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

   if -2.3499999999999999e-269 < x < 1.2500000000000001e91

   1. Initial program 65.2%

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

      1. fma-neg 64.8%

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

      2. /-rgt-identity 64.8%

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

      3. fma-neg 65.2%

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

      4. /-rgt-identity 65.2%

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

      5. distribute-rgt-neg-in 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around 0 46.8%

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

   6. Taylor expanded in eps around inf 80.4%

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

      1. +-commutative 80.4%

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

      2. associate-*r* 80.4%

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

      3. sub-neg 80.4%

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

      4. neg-mul-1 80.4%

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

      5. associate-*r* 80.4%

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

      6. neg-mul-1 80.4%

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

      7. distribute-lft-neg-in 80.4%

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

      8. *-commutative 80.4%

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

      9. associate-*r* 80.4%

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

      10. neg-mul-1 80.4%

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

      11. sub-neg 80.4%

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

      12. mul-1-neg 80.4%

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

   8. Simplified 80.4%

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

   9. Taylor expanded in eps around inf 80.7%

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

       1. *-commutative 80.7%

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

   11. Simplified 80.7%

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

   if 1.2500000000000001e91 < x < 8.0000000000000008e252 or 2.4000000000000002e273 < x

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 17.4%

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

   6. Taylor expanded in x around 0 68.3%

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

   if 8.0000000000000008e252 < x < 2.4000000000000002e273

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 14.8%

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

   6. Taylor expanded in x around 0 28.6%

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

      1. associate-*r* 28.6%

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

      2. sub-neg 28.6%

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

      3. metadata-eval 28.6%

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

      4. distribute-rgt-in 14.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 57.6%

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

      7. mul-1-neg 57.6%

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

      8. mul-1-neg 57.6%

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

      9. sqr-neg 57.6%

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

      10. sqrt-unprod 71.7%

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

      11. add-sqr-sqrt 71.7%

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

   8. Applied egg-rr 71.7%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-269}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+91}:\\ \;\;\;\;\frac{1 + \left(e^{\varepsilon \cdot x} - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+252} \lor \neg \left(x \leq 2.4 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)\right)\right) + \left(1 + \left(\left(x \cdot \left(1 + \varepsilon\right) + \frac{1}{\varepsilon} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + eps\_m\right)\\ t_1 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+252} \lor \neg \left(x \leq 10^{+274}\right):\\ \;\;\;\;\frac{t\_1 + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{eps\_m} + x \cdot \left(t\_1 \cdot \left(eps\_m + -1\right)\right)\right)\right) + \left(1 + \left(\left(t\_0 + \frac{1}{eps\_m} \cdot t\_0\right) + \frac{-1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 eps_m))) (t_1 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -1e-287)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 1.4e+93)
       (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
       (if (or (<= x 9e+252) (not (<= x 1e+274)))
         (/ (+ t_1 (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/
          (+
           (+ 1.0 (+ (/ 1.0 eps_m) (* x (* t_1 (+ eps_m -1.0)))))
           (+ 1.0 (+ (+ t_0 (* (/ 1.0 eps_m) t_0)) (/ -1.0 eps_m))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1e-287) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.4e+93) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else if ((x <= 9e+252) || !(x <= 1e+274)) {
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x * (1.0d0 + eps_m)
    t_1 = 1.0d0 + (1.0d0 / eps_m)
    if (x <= (-1d-287)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.4d+93) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else if ((x <= 9d+252) .or. (.not. (x <= 1d+274))) then
        tmp = (t_1 + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = ((1.0d0 + ((1.0d0 / eps_m) + (x * (t_1 * (eps_m + (-1.0d0)))))) + (1.0d0 + ((t_0 + ((1.0d0 / eps_m) * t_0)) + ((-1.0d0) / eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -1e-287) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.4e+93) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else if ((x <= 9e+252) || !(x <= 1e+274)) {
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + eps_m)
	t_1 = 1.0 + (1.0 / eps_m)
	tmp = 0
	if x <= -1e-287:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.4e+93:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	elif (x <= 9e+252) or not (x <= 1e+274):
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + eps_m))
	t_1 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -1e-287)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.4e+93)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	elseif ((x <= 9e+252) || !(x <= 1e+274))
		tmp = Float64(Float64(t_1 + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(x * Float64(t_1 * Float64(eps_m + -1.0))))) + Float64(1.0 + Float64(Float64(t_0 + Float64(Float64(1.0 / eps_m) * t_0)) + Float64(-1.0 / eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (1.0 + eps_m);
	t_1 = 1.0 + (1.0 / eps_m);
	tmp = 0.0;
	if (x <= -1e-287)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.4e+93)
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	elseif ((x <= 9e+252) || ~((x <= 1e+274)))
		tmp = (t_1 + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = ((1.0 + ((1.0 / eps_m) + (x * (t_1 * (eps_m + -1.0))))) + (1.0 + ((t_0 + ((1.0 / eps_m) * t_0)) + (-1.0 / eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-287], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4e+93], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 9e+252], N[Not[LessEqual[x, 1e+274]], $MachinePrecision]], N[(N[(t$95$1 + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * N[(t$95$1 * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(t$95$0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000002e-287

   1. Initial program 66.7%

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

      1. fma-neg 66.8%

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

      2. /-rgt-identity 66.8%

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

      3. fma-neg 66.7%

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

      4. /-rgt-identity 66.7%

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

      5. distribute-rgt-neg-in 66.7%

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

      6. sub-neg 66.7%

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

      7. metadata-eval 66.7%

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

      8. distribute-rgt-neg-in 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in x around 0 47.1%

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

   6. Taylor expanded in eps around inf 76.5%

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

      1. +-commutative 76.5%

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

      2. associate-*r* 76.5%

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

      3. sub-neg 76.5%

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

      4. neg-mul-1 76.5%

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

      5. associate-*r* 76.5%

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

      6. neg-mul-1 76.5%

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

      7. distribute-lft-neg-in 76.5%

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

      8. *-commutative 76.5%

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

      9. associate-*r* 76.5%

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

      10. neg-mul-1 76.5%

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

      11. sub-neg 76.5%

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

      12. mul-1-neg 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in eps around 0 79.4%

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

       1. mul-1-neg 79.4%

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

   11. Simplified 79.4%

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

   if -1.00000000000000002e-287 < x < 1.39999999999999994e93

   1. Initial program 65.9%

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

      1. fma-neg 65.5%

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

      2. /-rgt-identity 65.5%

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

      3. fma-neg 65.9%

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

      4. /-rgt-identity 65.9%

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

      5. distribute-rgt-neg-in 65.9%

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

      6. sub-neg 65.9%

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

      7. metadata-eval 65.9%

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

      8. distribute-rgt-neg-in 65.9%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in x around 0 46.2%

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

   6. Taylor expanded in eps around inf 79.6%

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

      1. associate-*r* 79.6%

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

      2. sub-neg 79.6%

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

      3. neg-mul-1 79.6%

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

      4. associate-*r* 79.6%

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

      5. associate-*r* 79.6%

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

      6. neg-mul-1 79.6%

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

      7. sub-neg 79.6%

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

      8. mul-1-neg 79.6%

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

   8. Simplified 79.6%

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

   if 1.39999999999999994e93 < x < 9.0000000000000001e252 or 9.99999999999999921e273 < x

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 17.4%

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

   6. Taylor expanded in x around 0 68.3%

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

   if 9.0000000000000001e252 < x < 9.99999999999999921e273

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 14.8%

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

   6. Taylor expanded in x around 0 28.6%

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

      1. associate-*r* 28.6%

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

      2. sub-neg 28.6%

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

      3. metadata-eval 28.6%

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

      4. distribute-rgt-in 14.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 57.6%

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

      7. mul-1-neg 57.6%

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

      8. mul-1-neg 57.6%

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

      9. sqr-neg 57.6%

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

      10. sqrt-unprod 71.7%

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

      11. add-sqr-sqrt 71.7%

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

   8. Applied egg-rr 71.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-287}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+93}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+252} \lor \neg \left(x \leq 10^{+274}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)\right)\right) + \left(1 + \left(\left(x \cdot \left(1 + \varepsilon\right) + \frac{1}{\varepsilon} \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right) + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1600:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + eps\_m\right) \cdot \left(\frac{1}{eps\_m} + -1\right) + \left(1 + \frac{1}{eps\_m}\right) \cdot \left(eps\_m + -1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1600.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/
    (*
     x
     (+
      (* (+ 1.0 eps_m) (+ (/ 1.0 eps_m) -1.0))
      (* (+ 1.0 (/ 1.0 eps_m)) (+ eps_m -1.0))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1600.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1600.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (x * (((1.0d0 + eps_m) * ((1.0d0 / eps_m) + (-1.0d0))) + ((1.0d0 + (1.0d0 / eps_m)) * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1600.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1600.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1600.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(1.0 + eps_m) * Float64(Float64(1.0 / eps_m) + -1.0)) + Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1600.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1600.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1600

   1. Initial program 63.1%

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

      1. fma-neg 62.9%

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

      2. /-rgt-identity 62.9%

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

      3. fma-neg 63.1%

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

      4. /-rgt-identity 63.1%

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

      5. distribute-rgt-neg-in 63.1%

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

      6. sub-neg 63.1%

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

      7. metadata-eval 63.1%

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

      8. distribute-rgt-neg-in 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 48.7%

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

   6. Taylor expanded in eps around inf 82.7%

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

      1. +-commutative 82.7%

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

      2. associate-*r* 82.7%

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

      3. sub-neg 82.7%

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

      4. neg-mul-1 82.7%

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

      5. associate-*r* 82.7%

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

      6. neg-mul-1 82.7%

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

      7. distribute-lft-neg-in 82.7%

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

      8. *-commutative 82.7%

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

      9. associate-*r* 82.7%

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

      10. neg-mul-1 82.7%

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

      11. sub-neg 82.7%

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

      12. mul-1-neg 82.7%

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

   8. Simplified 82.7%

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

   9. Taylor expanded in eps around 0 76.1%

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

       1. mul-1-neg 76.1%

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

   11. Simplified 76.1%

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

   if 1600 < x

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 23.0%

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

   6. Taylor expanded in x around 0 26.8%

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

   7. Taylor expanded in x around inf 57.3%

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

      1. distribute-lft-out-- 57.3%

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

      2. *-commutative 57.3%

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

      3. sub-neg 57.3%

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

      4. metadata-eval 57.3%

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

      5. +-commutative 57.3%

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

   9. Simplified 57.3%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1600:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + -1\right) + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 330:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + eps\_m\right) \cdot \left(\frac{1}{eps\_m} + -1\right) + \left(1 + \frac{1}{eps\_m}\right) \cdot \left(eps\_m + -1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 330.0)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (/
    (*
     x
     (+
      (* (+ 1.0 eps_m) (+ (/ 1.0 eps_m) -1.0))
      (* (+ 1.0 (/ 1.0 eps_m)) (+ eps_m -1.0))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 330.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 330.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else
        tmp = (x * (((1.0d0 + eps_m) * ((1.0d0 / eps_m) + (-1.0d0))) + ((1.0d0 + (1.0d0 / eps_m)) * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 330.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 330.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 330.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(1.0 + eps_m) * Float64(Float64(1.0 / eps_m) + -1.0)) + Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 330.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	else
		tmp = (x * (((1.0 + eps_m) * ((1.0 / eps_m) + -1.0)) + ((1.0 + (1.0 / eps_m)) * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 330.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 330

   1. Initial program 62.9%

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

      1. fma-neg 62.7%

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

      2. /-rgt-identity 62.7%

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

      3. fma-neg 62.9%

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

      4. /-rgt-identity 62.9%

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

      5. distribute-rgt-neg-in 62.9%

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

      6. sub-neg 62.9%

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

      7. metadata-eval 62.9%

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

      8. distribute-rgt-neg-in 62.9%

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

   3. Simplified 62.9%

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

   5. Taylor expanded in x around 0 48.4%

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

   6. Taylor expanded in x around 0 27.7%

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

   7. Taylor expanded in eps around inf 63.2%

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

      1. mul-1-neg 63.2%

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

      2. *-commutative 63.2%

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

      3. unsub-neg 63.2%

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

   9. Simplified 63.2%

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

   if 330 < x

   1. Initial program 100.0%

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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   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 24.0%

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

   6. Taylor expanded in x around 0 26.5%

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

   7. Taylor expanded in x around inf 56.6%

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

      1. distribute-lft-out-- 56.6%

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

      2. *-commutative 56.6%

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

      3. sub-neg 56.6%

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

      4. metadata-eval 56.6%

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

      5. +-commutative 56.6%

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

   9. Simplified 56.6%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 330:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} + -1\right) + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.25e-26)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.25e-26) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.25d-26) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.25e-26) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.25e-26:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.25e-26)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.25e-26)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.25e-26], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2499999999999999e-26

   1. Initial program 62.0%

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

      1. fma-neg 62.0%

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

      2. /-rgt-identity 62.0%

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

      3. fma-neg 62.0%

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

      4. /-rgt-identity 62.0%

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

      5. distribute-rgt-neg-in 62.0%

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

      6. sub-neg 62.0%

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

      7. metadata-eval 62.0%

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

      8. distribute-rgt-neg-in 62.0%

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

   3. Simplified 62.0%

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

   5. Taylor expanded in x around 0 48.2%

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

   6. Taylor expanded in x around 0 28.6%

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

   7. Taylor expanded in eps around inf 65.2%

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

      1. mul-1-neg 65.2%

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

      2. *-commutative 65.2%

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

      3. unsub-neg 65.2%

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

   9. Simplified 65.2%

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

   if 2.2499999999999999e-26 < x

   1. Initial program 98.3%

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

      1. fma-neg 97.8%

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

      2. /-rgt-identity 97.8%

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

      3. fma-neg 98.3%

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

      4. /-rgt-identity 98.3%

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

      5. distribute-rgt-neg-in 98.3%

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

      6. sub-neg 98.3%

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

      7. metadata-eval 98.3%

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

      8. distribute-rgt-neg-in 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in x around 0 27.0%

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

   6. Taylor expanded in x around 0 51.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0088:\\ \;\;\;\;\frac{eps\_m \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 72:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0088)
   (/ (* eps_m (- x)) 2.0)
   (if (<= x 72.0) 1.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0088) {
		tmp = (eps_m * -x) / 2.0;
	} else if (x <= 72.0) {
		tmp = 1.0;
	} 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 <= (-0.0088d0)) then
        tmp = (eps_m * -x) / 2.0d0
    else if (x <= 72.0d0) then
        tmp = 1.0d0
    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 <= -0.0088) {
		tmp = (eps_m * -x) / 2.0;
	} else if (x <= 72.0) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0088:
		tmp = (eps_m * -x) / 2.0
	elif x <= 72.0:
		tmp = 1.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0088)
		tmp = Float64(Float64(eps_m * Float64(-x)) / 2.0);
	elseif (x <= 72.0)
		tmp = 1.0;
	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 <= -0.0088)
		tmp = (eps_m * -x) / 2.0;
	elseif (x <= 72.0)
		tmp = 1.0;
	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, -0.0088], N[(N[(eps$95$m * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 72.0], 1.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00880000000000000053

   1. Initial program 94.5%

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

      1. fma-neg 94.6%

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

      2. /-rgt-identity 94.6%

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

      3. fma-neg 94.5%

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

      4. /-rgt-identity 94.5%

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

      5. distribute-rgt-neg-in 94.5%

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

      6. sub-neg 94.5%

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

      7. metadata-eval 94.5%

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

      8. distribute-rgt-neg-in 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around 0 59.7%

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

   6. Taylor expanded in x around 0 29.1%

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

   7. Taylor expanded in eps around inf 29.3%

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

      1. associate-*r* 29.3%

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

      2. mul-1-neg 29.3%

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

   9. Simplified 29.3%

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

   if -0.00880000000000000053 < x < 72

   1. Initial program 55.3%

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

      1. fma-neg 55.0%

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

      2. /-rgt-identity 55.0%

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

      3. fma-neg 55.3%

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

      4. /-rgt-identity 55.3%

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

      5. distribute-rgt-neg-in 55.3%

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

      6. sub-neg 55.3%

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

      7. metadata-eval 55.3%

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

      8. distribute-rgt-neg-in 55.3%

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

   3. Simplified 55.3%

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

   5. Taylor expanded in x around 0 73.6%

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

   if 72 < x

   1. Initial program 98.8%

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

      1. fma-neg 98.8%

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

      2. /-rgt-identity 98.8%

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

      3. fma-neg 98.8%

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

      4. /-rgt-identity 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. sub-neg 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-rgt-neg-in 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in eps around inf 98.9%

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

      1. exp-prod 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in x around 0 22.9%

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

      1. mul-1-neg 22.9%

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

      2. unsub-neg 22.9%

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

   10. Simplified 22.9%

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

   11. Taylor expanded in eps around inf 10.7%

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

       1. *-commutative 10.7%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   13. Simplified 10.7%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0088:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 72:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9:\\ \;\;\;\;\frac{2 - eps\_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9.0) (/ (- 2.0 (* eps_m x)) 2.0) (/ (* eps_m x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} 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 <= 9.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    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 <= 9.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	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 <= 9.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	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, 9.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9

   1. Initial program 63.2%

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

      1. fma-neg 63.0%

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

      2. /-rgt-identity 63.0%

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

      3. fma-neg 63.2%

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

      4. /-rgt-identity 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. sub-neg 63.2%

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

      7. metadata-eval 63.2%

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

      8. distribute-rgt-neg-in 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in x around 0 48.6%

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

   6. Taylor expanded in x around 0 27.8%

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

   7. Taylor expanded in eps around inf 63.6%

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

      1. mul-1-neg 63.6%

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

      2. *-commutative 63.6%

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

      3. unsub-neg 63.6%

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

   9. Simplified 63.6%

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

   if 9 < x

   1. Initial program 98.8%

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

      1. fma-neg 98.8%

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

      2. /-rgt-identity 98.8%

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

      3. fma-neg 98.8%

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

      4. /-rgt-identity 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. sub-neg 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-rgt-neg-in 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in eps around inf 98.9%

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

      1. exp-prod 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in x around 0 22.9%

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

      1. mul-1-neg 22.9%

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

      2. unsub-neg 22.9%

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

   10. Simplified 22.9%

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

   11. Taylor expanded in eps around inf 10.7%

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

       1. *-commutative 10.7%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   13. Simplified 10.7%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.2% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 64:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 64.0) 1.0 (/ (* eps_m x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 64.0) {
		tmp = 1.0;
	} 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 <= 64.0d0) then
        tmp = 1.0d0
    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 <= 64.0) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 64.0:
		tmp = 1.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 64.0)
		tmp = 1.0;
	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 <= 64.0)
		tmp = 1.0;
	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, 64.0], 1.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 64

   1. Initial program 63.2%

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

      1. fma-neg 63.0%

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

      2. /-rgt-identity 63.0%

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

      3. fma-neg 63.2%

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

      4. /-rgt-identity 63.2%

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

      5. distribute-rgt-neg-in 63.2%

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

      6. sub-neg 63.2%

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

      7. metadata-eval 63.2%

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

      8. distribute-rgt-neg-in 63.2%

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

   3. Simplified 63.2%

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

   5. Taylor expanded in x around 0 59.3%

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

   if 64 < x

   1. Initial program 98.8%

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

      1. fma-neg 98.8%

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

      2. /-rgt-identity 98.8%

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

      3. fma-neg 98.8%

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

      4. /-rgt-identity 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. sub-neg 98.8%

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

      7. metadata-eval 98.8%

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

      8. distribute-rgt-neg-in 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in eps around inf 98.9%

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

      1. exp-prod 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in x around 0 22.9%

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

      1. mul-1-neg 22.9%

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

      2. unsub-neg 22.9%

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

   10. Simplified 22.9%

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

   11. Taylor expanded in eps around inf 10.7%

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

       1. *-commutative 10.7%

          \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   13. Simplified 10.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 64:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.8% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. fma-neg 73.9%

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

   2. /-rgt-identity 73.9%

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

   3. fma-neg 74.1%

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

   4. /-rgt-identity 74.1%

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

   5. distribute-rgt-neg-in 74.1%

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

   6. sub-neg 74.1%

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

   7. metadata-eval 74.1%

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

   8. distribute-rgt-neg-in 74.1%

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

3. Simplified 74.1%

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

5. Taylor expanded in x around 0 42.2%

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

6. Final simplification 42.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(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))