NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 98.7%

Time: 21.4s

Alternatives: 15

Speedup: 1.9×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 4e-10)
     (/ (+ t_0 (exp (* x (- eps_m)))) 2.0)
     (/ (+ t_0 (exp (- x))) 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 <= 4e-10) {
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + exp(-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 * (eps_m + (-1.0d0))))
    if (x <= 4d-10) then
        tmp = (t_0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (t_0 + exp(-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 * (eps_m + -1.0)));
	double tmp;
	if (x <= 4e-10) {
		tmp = (t_0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (t_0 + Math.exp(-x)) / 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 <= 4e-10:
		tmp = (t_0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (t_0 + math.exp(-x)) / 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 <= 4e-10)
		tmp = Float64(Float64(t_0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(-x))) / 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 <= 4e-10)
		tmp = (t_0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = (t_0 + exp(-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[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4e-10], N[(N[(t$95$0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e-10

   1. Initial program 62.2%

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

      1. fma-neg 62.2%

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

      2. /-rgt-identity 62.2%

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

      3. fma-neg 62.2%

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

      4. /-rgt-identity 62.2%

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

      5. distribute-rgt-neg-in 62.2%

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

      6. sub-neg 62.2%

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

      7. metadata-eval 62.2%

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

      8. distribute-rgt-neg-in 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in eps around inf 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 -inf 98.9%

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

      1. associate-*r* 98.9%

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

      2. mul-1-neg 98.9%

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

      3. sub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. mul-1-neg 98.9%

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

      6. associate-*r* 98.9%

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

      7. mul-1-neg 98.9%

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

      8. mul-1-neg 98.9%

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

   8. Simplified 98.9%

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

   9. Taylor expanded in eps around inf 98.9%

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

   10. Taylor expanded in x around inf 98.9%

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

       1. associate-*r* 98.9%

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

       2. neg-mul-1 98.9%

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

       3. associate-*r* 98.9%

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

       4. neg-mul-1 98.9%

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

   12. Simplified 98.9%

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

   if 4.00000000000000015e-10 < x

   1. Initial program 99.1%

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

      1. fma-neg 99.1%

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

      2. /-rgt-identity 99.1%

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

      3. fma-neg 99.1%

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

      4. /-rgt-identity 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

      8. distribute-rgt-neg-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in eps around -inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. mul-1-neg 98.8%

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

      3. sub-neg 98.8%

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

      4. mul-1-neg 98.8%

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

      5. mul-1-neg 98.8%

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

      6. associate-*r* 98.8%

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

      7. mul-1-neg 98.8%

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

      8. mul-1-neg 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in eps around 0 65.6%

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

4. Final simplification 89.3%

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

Alternative 2: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-14)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-14) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -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 (eps_m <= 5d-14) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-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 (eps_m <= 5e-14) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -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 eps_m <= 5e-14:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -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 (eps_m <= 5e-14)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5e-14)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((x * (eps_m + -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[eps$95$m, 5e-14], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.0000000000000002e-14

   1. Initial program 60.3%

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

      1. fma-neg 60.2%

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

      2. /-rgt-identity 60.2%

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

      3. fma-neg 60.3%

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

      4. /-rgt-identity 60.3%

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

      5. distribute-rgt-neg-in 60.3%

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

      6. sub-neg 60.3%

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

      7. metadata-eval 60.3%

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

      8. distribute-rgt-neg-in 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around -inf 98.4%

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

      1. associate-*r* 98.4%

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

      2. mul-1-neg 98.4%

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

      3. sub-neg 98.4%

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

      4. mul-1-neg 98.4%

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

      5. mul-1-neg 98.4%

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

      6. associate-*r* 98.4%

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

      7. mul-1-neg 98.4%

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

      8. mul-1-neg 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in eps around 0 75.1%

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

   if 5.0000000000000002e-14 < 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%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. mul-1-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

   10. Taylor expanded in x around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

       3. associate-*r* 100.0%

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

       4. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. fma-neg 72.8%

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

   2. /-rgt-identity 72.8%

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

   3. fma-neg 72.8%

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

   4. /-rgt-identity 72.8%

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

   5. distribute-rgt-neg-in 72.8%

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

   6. sub-neg 72.8%

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

   7. metadata-eval 72.8%

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

   8. distribute-rgt-neg-in 72.8%

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

3. Simplified 72.8%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2}} \]
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 -inf 98.9%

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

   1. associate-*r* 98.9%

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

   2. mul-1-neg 98.9%

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

   3. sub-neg 98.9%

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

   4. mul-1-neg 98.9%

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

   5. mul-1-neg 98.9%

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

   6. associate-*r* 98.9%

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

   7. mul-1-neg 98.9%

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

   8. mul-1-neg 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 4: 85.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\left(x + \left(e^{x \cdot \left(eps\_m + -1\right)} - x \cdot eps\_m\right)\right) + \left(1 - x\right)}{2}\\ t_1 := \left(x \cdot eps\_m\right) \cdot 0.5\\ \mathbf{if}\;x \leq -6 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.66 \cdot 10^{+183}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+225}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (+ (+ x (- (exp (* x (+ eps_m -1.0))) (* x eps_m))) (- 1.0 x))
          2.0))
        (t_1 (* (* x eps_m) 0.5)))
   (if (<= x -6e-257)
     (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
     (if (<= x 4.2e+94)
       t_0
       (if (<= x 2.8e+155)
         t_1
         (if (<= x 1.66e+183)
           t_0
           (if (<= x 6e+225) t_1 (* 0.25 (/ (* x x) eps_m)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((x + (exp((x * (eps_m + -1.0))) - (x * eps_m))) + (1.0 - x)) / 2.0;
	double t_1 = (x * eps_m) * 0.5;
	double tmp;
	if (x <= -6e-257) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 4.2e+94) {
		tmp = t_0;
	} else if (x <= 2.8e+155) {
		tmp = t_1;
	} else if (x <= 1.66e+183) {
		tmp = t_0;
	} else if (x <= 6e+225) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x + (exp((x * (eps_m + (-1.0d0)))) - (x * eps_m))) + (1.0d0 - x)) / 2.0d0
    t_1 = (x * eps_m) * 0.5d0
    if (x <= (-6d-257)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 4.2d+94) then
        tmp = t_0
    else if (x <= 2.8d+155) then
        tmp = t_1
    else if (x <= 1.66d+183) then
        tmp = t_0
    else if (x <= 6d+225) then
        tmp = t_1
    else
        tmp = 0.25d0 * ((x * x) / eps_m)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((x + (Math.exp((x * (eps_m + -1.0))) - (x * eps_m))) + (1.0 - x)) / 2.0;
	double t_1 = (x * eps_m) * 0.5;
	double tmp;
	if (x <= -6e-257) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 4.2e+94) {
		tmp = t_0;
	} else if (x <= 2.8e+155) {
		tmp = t_1;
	} else if (x <= 1.66e+183) {
		tmp = t_0;
	} else if (x <= 6e+225) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = ((x + (math.exp((x * (eps_m + -1.0))) - (x * eps_m))) + (1.0 - x)) / 2.0
	t_1 = (x * eps_m) * 0.5
	tmp = 0
	if x <= -6e-257:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 4.2e+94:
		tmp = t_0
	elif x <= 2.8e+155:
		tmp = t_1
	elif x <= 1.66e+183:
		tmp = t_0
	elif x <= 6e+225:
		tmp = t_1
	else:
		tmp = 0.25 * ((x * x) / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(x + Float64(exp(Float64(x * Float64(eps_m + -1.0))) - Float64(x * eps_m))) + Float64(1.0 - x)) / 2.0)
	t_1 = Float64(Float64(x * eps_m) * 0.5)
	tmp = 0.0
	if (x <= -6e-257)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 4.2e+94)
		tmp = t_0;
	elseif (x <= 2.8e+155)
		tmp = t_1;
	elseif (x <= 1.66e+183)
		tmp = t_0;
	elseif (x <= 6e+225)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(x * x) / eps_m));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((x + (exp((x * (eps_m + -1.0))) - (x * eps_m))) + (1.0 - x)) / 2.0;
	t_1 = (x * eps_m) * 0.5;
	tmp = 0.0;
	if (x <= -6e-257)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 4.2e+94)
		tmp = t_0;
	elseif (x <= 2.8e+155)
		tmp = t_1;
	elseif (x <= 1.66e+183)
		tmp = t_0;
	elseif (x <= 6e+225)
		tmp = t_1;
	else
		tmp = 0.25 * ((x * x) / eps_m);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(x + N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -6e-257], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2e+94], t$95$0, If[LessEqual[x, 2.8e+155], t$95$1, If[LessEqual[x, 1.66e+183], t$95$0, If[LessEqual[x, 6e+225], t$95$1, N[(0.25 * N[(N[(x * x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.9999999999999999e-257

   1. Initial program 69.8%

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

      1. fma-neg 69.7%

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

      2. /-rgt-identity 69.7%

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

      3. fma-neg 69.8%

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

      4. /-rgt-identity 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. distribute-rgt-neg-in 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around -inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. mul-1-neg 98.0%

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

      5. mul-1-neg 98.0%

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

      6. associate-*r* 98.0%

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

      7. mul-1-neg 98.0%

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

      8. mul-1-neg 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in eps around inf 98.0%

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

   10. Taylor expanded in x around 0 76.4%

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

   if -5.9999999999999999e-257 < x < 4.19999999999999979e94 or 2.80000000000000016e155 < x < 1.66000000000000008e183

   1. Initial program 64.6%

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

      1. fma-neg 64.6%

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

      2. /-rgt-identity 64.6%

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

      3. fma-neg 64.6%

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

      4. /-rgt-identity 64.6%

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

      5. distribute-rgt-neg-in 64.6%

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

      6. sub-neg 64.6%

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

      7. metadata-eval 64.6%

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

      8. distribute-rgt-neg-in 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in x around 0 39.2%

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

   6. Taylor expanded in eps around inf 74.2%

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

   8. Simplified 85.6%

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

   if 4.19999999999999979e94 < x < 2.80000000000000016e155 or 1.66000000000000008e183 < x < 6.000000000000001e225

   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 38.7%

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

   6. Taylor expanded in eps around inf 39.0%

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

      1. associate-*r* 39.0%

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

      2. mul-1-neg 39.0%

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

   8. Simplified 39.0%

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

      1. div-inv 39.0%

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

      2. add-sqr-sqrt 39.0%

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

      3. sqrt-unprod 38.8%

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

      4. sqr-neg 38.8%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 29.5%

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

      7. distribute-rgt-neg-in 29.5%

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

      8. *-commutative 29.5%

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

      9. distribute-rgt-neg-in 29.5%

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

      10. remove-double-neg 29.5%

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

      11. metadata-eval 29.5%

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

   10. Applied egg-rr 29.5%

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

   if 6.000000000000001e225 < 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 48.6%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]

   9. Taylor expanded in x around 0 20.6%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. unpow2 20.6%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]

   11. Applied egg-rr 20.6%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\left(x + \left(e^{x \cdot \left(\varepsilon + -1\right)} - x \cdot \varepsilon\right)\right) + \left(1 - x\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+155}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.66 \cdot 10^{+183}:\\ \;\;\;\;\frac{\left(x + \left(e^{x \cdot \left(\varepsilon + -1\right)} - x \cdot \varepsilon\right)\right) + \left(1 - x\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+225}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 420:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 420.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 9e+85)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
     (* 0.5 (/ (+ x (expm1 x)) eps_m)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 420.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 9e+85) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = 0.5 * ((x + expm1(x)) / eps_m);
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 420.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 9e+85) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = 0.5 * ((x + Math.expm1(x)) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 420.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 9e+85:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = 0.5 * ((x + math.expm1(x)) / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 420.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 9e+85)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(0.5 * Float64(Float64(x + expm1(x)) / eps_m));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 420.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+85], 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], N[(0.5 * N[(N[(x + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 420

   1. Initial program 62.8%

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

      1. fma-neg 62.8%

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

      2. /-rgt-identity 62.8%

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

      3. fma-neg 62.8%

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

      4. /-rgt-identity 62.8%

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

      5. distribute-rgt-neg-in 62.8%

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

      6. sub-neg 62.8%

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

      7. metadata-eval 62.8%

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

      8. distribute-rgt-neg-in 62.8%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around -inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. mul-1-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. mul-1-neg 98.5%

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

      5. mul-1-neg 98.5%

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

      6. associate-*r* 98.5%

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

      7. mul-1-neg 98.5%

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

      8. mul-1-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in eps around inf 98.4%

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

   10. Taylor expanded in eps around 0 76.4%

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

       1. neg-mul-1 76.4%

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

   12. Simplified 76.4%

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

   if 420 < x < 9.00000000000000013e85

   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 14.4%

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

   6. Taylor expanded in x around 0 75.8%

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

   if 9.00000000000000013e85 < 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 40.5%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon} \cdot \frac{1}{2}} \]

      2. +-commutative 1.6%

         \[\leadsto \frac{\color{blue}{x + \mathsf{expm1}\left(-x\right)}}{\varepsilon} \cdot \frac{1}{2} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      4. sqrt-unprod 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      5. sqr-neg 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      6. sqrt-unprod 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      7. add-sqr-sqrt 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      8. metadata-eval 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon} \cdot \color{blue}{0.5} \]

   10. Applied egg-rr 23.2%

       \[\leadsto \color{blue}{\frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 420:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+85}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+90}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-259)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 1e+90)
     (/ (* 2.0 (exp (- x))) 2.0)
     (* 0.5 (/ (+ x (expm1 x)) eps_m)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-259) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 1e+90) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = 0.5 * ((x + expm1(x)) / eps_m);
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-259) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 1e+90) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.5 * ((x + Math.expm1(x)) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-259:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 1e+90:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = 0.5 * ((x + math.expm1(x)) / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-259)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 1e+90)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(0.5 * Float64(Float64(x + expm1(x)) / eps_m));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-259], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+90], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(N[(x + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-259

   1. Initial program 69.8%

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

      1. fma-neg 69.7%

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

      2. /-rgt-identity 69.7%

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

      3. fma-neg 69.8%

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

      4. /-rgt-identity 69.8%

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

      5. distribute-rgt-neg-in 69.8%

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

      6. sub-neg 69.8%

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

      7. metadata-eval 69.8%

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

      8. distribute-rgt-neg-in 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around -inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. mul-1-neg 98.0%

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

      3. sub-neg 98.0%

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

      4. mul-1-neg 98.0%

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

      5. mul-1-neg 98.0%

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

      6. associate-*r* 98.0%

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

      7. mul-1-neg 98.0%

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

      8. mul-1-neg 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in eps around inf 98.0%

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

   10. Taylor expanded in x around 0 76.4%

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

   if -2.0000000000000001e-259 < x < 9.99999999999999966e89

   1. Initial program 62.4%

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

      1. fma-neg 62.3%

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

      2. /-rgt-identity 62.3%

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

      3. fma-neg 62.4%

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

      4. /-rgt-identity 62.4%

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

      5. distribute-rgt-neg-in 62.4%

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

      6. sub-neg 62.4%

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

      7. metadata-eval 62.4%

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

      8. distribute-rgt-neg-in 62.4%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around -inf 99.1%

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

      1. associate-*r* 99.1%

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

      2. mul-1-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. mul-1-neg 99.1%

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

      5. mul-1-neg 99.1%

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

      6. associate-*r* 99.1%

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

      7. mul-1-neg 99.1%

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

      8. mul-1-neg 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in eps around 0 74.7%

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

   if 9.99999999999999966e89 < 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 40.5%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon} \cdot \frac{1}{2}} \]

      2. +-commutative 1.6%

         \[\leadsto \frac{\color{blue}{x + \mathsf{expm1}\left(-x\right)}}{\varepsilon} \cdot \frac{1}{2} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      4. sqrt-unprod 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      5. sqr-neg 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      6. sqrt-unprod 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      7. add-sqr-sqrt 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      8. metadata-eval 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon} \cdot \color{blue}{0.5} \]

   10. Applied egg-rr 23.2%

       \[\leadsto \color{blue}{\frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon} \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+90}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+90}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1e+90)
   (/ (* 2.0 (exp (- x))) 2.0)
   (* 0.5 (/ (+ x (expm1 x)) eps_m))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1e+90) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = 0.5 * ((x + expm1(x)) / eps_m);
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1e+90) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.5 * ((x + Math.expm1(x)) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1e+90:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = 0.5 * ((x + math.expm1(x)) / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1e+90)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(0.5 * Float64(Float64(x + expm1(x)) / eps_m));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1e+90], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(N[(x + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999966e89

   1. Initial program 65.8%

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

      1. fma-neg 65.7%

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

      2. /-rgt-identity 65.7%

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

      3. fma-neg 65.8%

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

      4. /-rgt-identity 65.8%

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

      5. distribute-rgt-neg-in 65.8%

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

      6. sub-neg 65.8%

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

      7. metadata-eval 65.8%

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

      8. distribute-rgt-neg-in 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in eps around -inf 98.6%

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

      1. associate-*r* 98.6%

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

      2. mul-1-neg 98.6%

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

      3. sub-neg 98.6%

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

      4. mul-1-neg 98.6%

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

      5. mul-1-neg 98.6%

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

      6. associate-*r* 98.6%

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

      7. mul-1-neg 98.6%

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

      8. mul-1-neg 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in eps around 0 76.3%

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

   if 9.99999999999999966e89 < 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 40.5%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. div-inv 1.6%

         \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon} \cdot \frac{1}{2}} \]

      2. +-commutative 1.6%

         \[\leadsto \frac{\color{blue}{x + \mathsf{expm1}\left(-x\right)}}{\varepsilon} \cdot \frac{1}{2} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      4. sqrt-unprod 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      5. sqr-neg 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      6. sqrt-unprod 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      7. add-sqr-sqrt 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon} \cdot \frac{1}{2} \]

      8. metadata-eval 23.2%

         \[\leadsto \frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon} \cdot \color{blue}{0.5} \]

   10. Applied egg-rr 23.2%

       \[\leadsto \color{blue}{\frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+90}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x + \mathsf{expm1}\left(x\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{if}\;x \leq 420:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+155}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)))
   (if (<= x 420.0)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 5e+87)
       t_0
       (if (<= x 3e+155)
         (* (* x eps_m) 0.5)
         (if (<= x 5.5e+179) t_0 (* 0.25 (/ (* x x) eps_m))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= 420.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 5e+87) {
		tmp = t_0;
	} else if (x <= 3e+155) {
		tmp = (x * eps_m) * 0.5;
	} else if (x <= 5.5e+179) {
		tmp = t_0;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    if (x <= 420.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 5d+87) then
        tmp = t_0
    else if (x <= 3d+155) then
        tmp = (x * eps_m) * 0.5d0
    else if (x <= 5.5d+179) then
        tmp = t_0
    else
        tmp = 0.25d0 * ((x * x) / eps_m)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	double tmp;
	if (x <= 420.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 5e+87) {
		tmp = t_0;
	} else if (x <= 3e+155) {
		tmp = (x * eps_m) * 0.5;
	} else if (x <= 5.5e+179) {
		tmp = t_0;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	tmp = 0
	if x <= 420.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 5e+87:
		tmp = t_0
	elif x <= 3e+155:
		tmp = (x * eps_m) * 0.5
	elif x <= 5.5e+179:
		tmp = t_0
	else:
		tmp = 0.25 * ((x * x) / eps_m)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0)
	tmp = 0.0
	if (x <= 420.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 5e+87)
		tmp = t_0;
	elseif (x <= 3e+155)
		tmp = Float64(Float64(x * eps_m) * 0.5);
	elseif (x <= 5.5e+179)
		tmp = t_0;
	else
		tmp = Float64(0.25 * Float64(Float64(x * x) / eps_m));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	tmp = 0.0;
	if (x <= 420.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 5e+87)
		tmp = t_0;
	elseif (x <= 3e+155)
		tmp = (x * eps_m) * 0.5;
	elseif (x <= 5.5e+179)
		tmp = t_0;
	else
		tmp = 0.25 * ((x * x) / eps_m);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = 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]}, If[LessEqual[x, 420.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+87], t$95$0, If[LessEqual[x, 3e+155], N[(N[(x * eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.5e+179], t$95$0, N[(0.25 * N[(N[(x * x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 420

   1. Initial program 62.8%

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

      1. fma-neg 62.8%

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

      2. /-rgt-identity 62.8%

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

      3. fma-neg 62.8%

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

      4. /-rgt-identity 62.8%

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

      5. distribute-rgt-neg-in 62.8%

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

      6. sub-neg 62.8%

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

      7. metadata-eval 62.8%

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

      8. distribute-rgt-neg-in 62.8%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around -inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. mul-1-neg 98.5%

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

      3. sub-neg 98.5%

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

      4. mul-1-neg 98.5%

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

      5. mul-1-neg 98.5%

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

      6. associate-*r* 98.5%

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

      7. mul-1-neg 98.5%

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

      8. mul-1-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in eps around inf 98.4%

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

   10. Taylor expanded in eps around 0 76.4%

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

       1. neg-mul-1 76.4%

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

   12. Simplified 76.4%

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

   if 420 < x < 4.9999999999999998e87 or 3.0000000000000001e155 < x < 5.4999999999999998e179

   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 16.2%

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

   6. Taylor expanded in x around 0 76.9%

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

   if 4.9999999999999998e87 < x < 3.0000000000000001e155

   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 28.3%

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

   6. Taylor expanded in eps around inf 28.5%

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

      1. associate-*r* 28.5%

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

      2. mul-1-neg 28.5%

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

   8. Simplified 28.5%

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

      1. div-inv 28.5%

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

      2. add-sqr-sqrt 28.5%

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

      3. sqrt-unprod 28.5%

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

      4. sqr-neg 28.5%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 37.1%

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

      7. distribute-rgt-neg-in 37.1%

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

      8. *-commutative 37.1%

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

      9. distribute-rgt-neg-in 37.1%

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

      10. remove-double-neg 37.1%

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

      11. metadata-eval 37.1%

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

   10. Applied egg-rr 37.1%

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

   if 5.4999999999999998e179 < 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 49.2%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. unpow2 22.2%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]

   11. Applied egg-rr 22.2%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 420:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+155}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+179}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{-1}{eps\_m}\\ t_1 := \frac{\left(1 + \frac{1}{eps\_m}\right) + t\_0}{2}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(t\_0 \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+155}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -1.0 eps_m)))
        (t_1 (/ (+ (+ 1.0 (/ 1.0 eps_m)) t_0) 2.0)))
   (if (<= x -1.0)
     (/ (* x (* t_0 (- -1.0 eps_m))) 2.0)
     (if (<= x 4e-10)
       1.0
       (if (<= x 4e+91)
         t_1
         (if (<= x 3.3e+155)
           (* (* x eps_m) 0.5)
           (if (<= x 5.2e+179) t_1 (* 0.25 (/ (* x x) eps_m)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (-1.0 / eps_m);
	double t_1 = ((1.0 + (1.0 / eps_m)) + t_0) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 4e-10) {
		tmp = 1.0;
	} else if (x <= 4e+91) {
		tmp = t_1;
	} else if (x <= 3.3e+155) {
		tmp = (x * eps_m) * 0.5;
	} else if (x <= 5.2e+179) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((-1.0d0) / eps_m)
    t_1 = ((1.0d0 + (1.0d0 / eps_m)) + t_0) / 2.0d0
    if (x <= (-1.0d0)) then
        tmp = (x * (t_0 * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 4d-10) then
        tmp = 1.0d0
    else if (x <= 4d+91) then
        tmp = t_1
    else if (x <= 3.3d+155) then
        tmp = (x * eps_m) * 0.5d0
    else if (x <= 5.2d+179) then
        tmp = t_1
    else
        tmp = 0.25d0 * ((x * x) / eps_m)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (-1.0 / eps_m);
	double t_1 = ((1.0 + (1.0 / eps_m)) + t_0) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 4e-10) {
		tmp = 1.0;
	} else if (x <= 4e+91) {
		tmp = t_1;
	} else if (x <= 3.3e+155) {
		tmp = (x * eps_m) * 0.5;
	} else if (x <= 5.2e+179) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (-1.0 / eps_m)
	t_1 = ((1.0 + (1.0 / eps_m)) + t_0) / 2.0
	tmp = 0
	if x <= -1.0:
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0
	elif x <= 4e-10:
		tmp = 1.0
	elif x <= 4e+91:
		tmp = t_1
	elif x <= 3.3e+155:
		tmp = (x * eps_m) * 0.5
	elif x <= 5.2e+179:
		tmp = t_1
	else:
		tmp = 0.25 * ((x * x) / eps_m)
	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(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + t_0) / 2.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(t_0 * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 4e-10)
		tmp = 1.0;
	elseif (x <= 4e+91)
		tmp = t_1;
	elseif (x <= 3.3e+155)
		tmp = Float64(Float64(x * eps_m) * 0.5);
	elseif (x <= 5.2e+179)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(x * x) / eps_m));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (-1.0 / eps_m);
	t_1 = ((1.0 + (1.0 / eps_m)) + t_0) / 2.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 4e-10)
		tmp = 1.0;
	elseif (x <= 4e+91)
		tmp = t_1;
	elseif (x <= 3.3e+155)
		tmp = (x * eps_m) * 0.5;
	elseif (x <= 5.2e+179)
		tmp = t_1;
	else
		tmp = 0.25 * ((x * x) / eps_m);
	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[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(x * N[(t$95$0 * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e-10], 1.0, If[LessEqual[x, 4e+91], t$95$1, If[LessEqual[x, 3.3e+155], N[(N[(x * eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e+179], t$95$1, N[(0.25 * N[(N[(x * x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 73.8%

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

   6. Taylor expanded in x around inf 49.5%

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

   if -1 < x < 4.00000000000000015e-10

   1. Initial program 53.2%

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

      1. fma-neg 53.2%

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

      2. /-rgt-identity 53.2%

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

      3. fma-neg 53.2%

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

      4. /-rgt-identity 53.2%

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

      5. distribute-rgt-neg-in 53.2%

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

      6. sub-neg 53.2%

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

      7. metadata-eval 53.2%

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

      8. distribute-rgt-neg-in 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 4.00000000000000015e-10 < x < 4.00000000000000032e91 or 3.2999999999999999e155 < x < 5.2000000000000004e179

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 21.2%

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

   6. Taylor expanded in x around 0 62.8%

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

   if 4.00000000000000032e91 < x < 3.2999999999999999e155

   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 28.3%

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

   6. Taylor expanded in eps around inf 28.5%

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

      1. associate-*r* 28.5%

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

      2. mul-1-neg 28.5%

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

   8. Simplified 28.5%

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

      1. div-inv 28.5%

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

      2. add-sqr-sqrt 28.5%

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

      3. sqrt-unprod 28.5%

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

      4. sqr-neg 28.5%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 37.1%

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

      7. distribute-rgt-neg-in 37.1%

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

      8. *-commutative 37.1%

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

      9. distribute-rgt-neg-in 37.1%

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

      10. remove-double-neg 37.1%

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

      11. metadata-eval 37.1%

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

   10. Applied egg-rr 37.1%

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

   if 5.2000000000000004e179 < 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 49.2%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. unpow2 22.2%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]

   11. Applied egg-rr 22.2%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{-1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+155}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{-1}{eps\_m}\\ t_1 := \frac{\frac{1}{eps\_m} + t\_0}{2}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(t\_0 \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+156}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{eps\_m}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -1.0 eps_m))) (t_1 (/ (+ (/ 1.0 eps_m) t_0) 2.0)))
   (if (<= x -1.0)
     (/ (* x (* t_0 (- -1.0 eps_m))) 2.0)
     (if (<= x 360.0)
       1.0
       (if (<= x 7e+92)
         t_1
         (if (<= x 3.7e+156)
           (* (* x eps_m) 0.5)
           (if (<= x 6e+179) t_1 (* 0.25 (/ (* x x) eps_m)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (-1.0 / eps_m);
	double t_1 = ((1.0 / eps_m) + t_0) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 7e+92) {
		tmp = t_1;
	} else if (x <= 3.7e+156) {
		tmp = (x * eps_m) * 0.5;
	} else if (x <= 6e+179) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((-1.0d0) / eps_m)
    t_1 = ((1.0d0 / eps_m) + t_0) / 2.0d0
    if (x <= (-1.0d0)) then
        tmp = (x * (t_0 * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if (x <= 7d+92) then
        tmp = t_1
    else if (x <= 3.7d+156) then
        tmp = (x * eps_m) * 0.5d0
    else if (x <= 6d+179) then
        tmp = t_1
    else
        tmp = 0.25d0 * ((x * x) / eps_m)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (-1.0 / eps_m);
	double t_1 = ((1.0 / eps_m) + t_0) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 7e+92) {
		tmp = t_1;
	} else if (x <= 3.7e+156) {
		tmp = (x * eps_m) * 0.5;
	} else if (x <= 6e+179) {
		tmp = t_1;
	} else {
		tmp = 0.25 * ((x * x) / eps_m);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (-1.0 / eps_m)
	t_1 = ((1.0 / eps_m) + t_0) / 2.0
	tmp = 0
	if x <= -1.0:
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0
	elif x <= 360.0:
		tmp = 1.0
	elif x <= 7e+92:
		tmp = t_1
	elif x <= 3.7e+156:
		tmp = (x * eps_m) * 0.5
	elif x <= 6e+179:
		tmp = t_1
	else:
		tmp = 0.25 * ((x * x) / eps_m)
	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(Float64(Float64(1.0 / eps_m) + t_0) / 2.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(t_0 * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 7e+92)
		tmp = t_1;
	elseif (x <= 3.7e+156)
		tmp = Float64(Float64(x * eps_m) * 0.5);
	elseif (x <= 6e+179)
		tmp = t_1;
	else
		tmp = Float64(0.25 * Float64(Float64(x * x) / eps_m));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (-1.0 / eps_m);
	t_1 = ((1.0 / eps_m) + t_0) / 2.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * (t_0 * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 7e+92)
		tmp = t_1;
	elseif (x <= 3.7e+156)
		tmp = (x * eps_m) * 0.5;
	elseif (x <= 6e+179)
		tmp = t_1;
	else
		tmp = 0.25 * ((x * x) / eps_m);
	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[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(x * N[(t$95$0 * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[LessEqual[x, 7e+92], t$95$1, If[LessEqual[x, 3.7e+156], N[(N[(x * eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6e+179], t$95$1, N[(0.25 * N[(N[(x * x), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 73.8%

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

   6. Taylor expanded in x around inf 49.5%

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

   if -1 < x < 360

   1. Initial program 54.5%

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

      1. fma-neg 54.4%

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

      2. /-rgt-identity 54.4%

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

      3. fma-neg 54.5%

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

      4. /-rgt-identity 54.5%

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

      5. distribute-rgt-neg-in 54.5%

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

      6. sub-neg 54.5%

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

      7. metadata-eval 54.5%

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

      8. distribute-rgt-neg-in 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in x around 0 72.0%

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

   if 360 < x < 6.99999999999999972e92 or 3.70000000000000001e156 < x < 5.9999999999999996e179

   1. Initial program 96.8%

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

      1. fma-neg 96.8%

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

      2. /-rgt-identity 96.8%

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

      3. fma-neg 96.8%

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

      4. /-rgt-identity 96.8%

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

      5. distribute-rgt-neg-in 96.8%

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

      6. sub-neg 96.8%

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

      7. metadata-eval 96.8%

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

      8. distribute-rgt-neg-in 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in x around 0 15.6%

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

   6. Taylor expanded in x around 0 73.6%

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

   7. Taylor expanded in eps around 0 73.6%

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

   if 6.99999999999999972e92 < x < 3.70000000000000001e156

   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 28.3%

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

   6. Taylor expanded in eps around inf 28.5%

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

      1. associate-*r* 28.5%

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

      2. mul-1-neg 28.5%

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

   8. Simplified 28.5%

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

      1. div-inv 28.5%

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

      2. add-sqr-sqrt 28.5%

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

      3. sqrt-unprod 28.5%

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

      4. sqr-neg 28.5%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 37.1%

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

      7. distribute-rgt-neg-in 37.1%

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

      8. *-commutative 37.1%

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

      9. distribute-rgt-neg-in 37.1%

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

      10. remove-double-neg 37.1%

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

      11. metadata-eval 37.1%

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

   10. Applied egg-rr 37.1%

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

   if 5.9999999999999996e179 < 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 49.2%

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

   6. Taylor expanded in eps around 0 1.6%

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

      1. associate--r+ 1.6%

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

      2. cancel-sign-sub-inv 1.6%

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

      3. expm1-def 1.6%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)} + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      4. mul-1-neg 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right) + \left(--1\right) \cdot x}{\varepsilon}}{2} \]

      5. metadata-eval 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{1} \cdot x}{\varepsilon}}{2} \]

      6. *-lft-identity 1.6%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(-x\right) + \color{blue}{x}}{\varepsilon}}{2} \]

   8. Simplified 1.6%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right) + x}{\varepsilon}}}{2} \]

   9. Taylor expanded in x around 0 22.2%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   10. Step-by-step derivation

       1. unpow2 22.2%

          \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]

   11. Applied egg-rr 22.2%

       \[\leadsto 0.25 \cdot \frac{\color{blue}{x \cdot x}}{\varepsilon} \]
3. Recombined 5 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{-1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+156}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{x \cdot x}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{-1}{eps\_m}\right) \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (/ (* x (* (+ 1.0 (/ -1.0 eps_m)) (- -1.0 eps_m))) 2.0)
   (if (<= x 4e-10) 1.0 (* (* x eps_m) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * ((1.0 + (-1.0 / eps_m)) * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 4e-10) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * ((1.0d0 + ((-1.0d0) / eps_m)) * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 4d-10) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * ((1.0 + (-1.0 / eps_m)) * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 4e-10) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * ((1.0 + (-1.0 / eps_m)) * (-1.0 - eps_m))) / 2.0
	elif x <= 4e-10:
		tmp = 1.0
	else:
		tmp = (x * eps_m) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(Float64(1.0 + Float64(-1.0 / eps_m)) * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 4e-10)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * ((1.0 + (-1.0 / eps_m)) * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 4e-10)
		tmp = 1.0;
	else
		tmp = (x * eps_m) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e-10], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 73.8%

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

   6. Taylor expanded in x around inf 49.5%

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

   if -1 < x < 4.00000000000000015e-10

   1. Initial program 53.2%

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

      1. fma-neg 53.2%

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

      2. /-rgt-identity 53.2%

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

      3. fma-neg 53.2%

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

      4. /-rgt-identity 53.2%

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

      5. distribute-rgt-neg-in 53.2%

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

      6. sub-neg 53.2%

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

      7. metadata-eval 53.2%

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

      8. distribute-rgt-neg-in 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 4.00000000000000015e-10 < x

   1. Initial program 99.1%

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

      1. fma-neg 99.1%

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

      2. /-rgt-identity 99.1%

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

      3. fma-neg 99.1%

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

      4. /-rgt-identity 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

      8. distribute-rgt-neg-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 35.1%

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

   6. Taylor expanded in eps around inf 27.8%

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

      1. associate-*r* 27.8%

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

      2. mul-1-neg 27.8%

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

   8. Simplified 27.8%

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

      1. div-inv 27.8%

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

      2. add-sqr-sqrt 27.8%

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

      3. sqrt-unprod 28.9%

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

      4. sqr-neg 28.9%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 15.6%

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

      7. distribute-rgt-neg-in 15.6%

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

      8. *-commutative 15.6%

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

      9. distribute-rgt-neg-in 15.6%

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

      10. remove-double-neg 15.6%

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

      11. metadata-eval 15.6%

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

   10. Applied egg-rr 15.6%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{-1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.4% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* x eps_m) -0.5)
   (if (<= x 4e-10) 1.0 (* (* x eps_m) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 4e-10) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 4d-10) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 4e-10) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) * -0.5
	elif x <= 4e-10:
		tmp = 1.0
	else:
		tmp = (x * eps_m) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 4e-10)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 4e-10)
		tmp = 1.0;
	else
		tmp = (x * eps_m) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 4e-10], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 73.8%

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

   6. Taylor expanded in eps around inf 49.6%

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

      1. associate-*r* 49.6%

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

      2. mul-1-neg 49.6%

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

   8. Simplified 49.6%

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

      1. frac-2neg 49.6%

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

      2. div-inv 49.6%

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

      3. *-commutative 49.6%

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

      4. distribute-rgt-neg-in 49.6%

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

      5. remove-double-neg 49.6%

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

      6. metadata-eval 49.6%

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

      7. metadata-eval 49.6%

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

   10. Applied egg-rr 49.6%

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

   if -1 < x < 4.00000000000000015e-10

   1. Initial program 53.2%

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

      1. fma-neg 53.2%

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

      2. /-rgt-identity 53.2%

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

      3. fma-neg 53.2%

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

      4. /-rgt-identity 53.2%

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

      5. distribute-rgt-neg-in 53.2%

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

      6. sub-neg 53.2%

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

      7. metadata-eval 53.2%

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

      8. distribute-rgt-neg-in 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in x around 0 73.9%

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

   if 4.00000000000000015e-10 < x

   1. Initial program 99.1%

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

      1. fma-neg 99.1%

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

      2. /-rgt-identity 99.1%

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

      3. fma-neg 99.1%

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

      4. /-rgt-identity 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. sub-neg 99.1%

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

      7. metadata-eval 99.1%

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

      8. distribute-rgt-neg-in 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in x around 0 35.1%

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

   6. Taylor expanded in eps around inf 27.8%

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

      1. associate-*r* 27.8%

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

      2. mul-1-neg 27.8%

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

   8. Simplified 27.8%

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

      1. div-inv 27.8%

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

      2. add-sqr-sqrt 27.8%

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

      3. sqrt-unprod 28.9%

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

      4. sqr-neg 28.9%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 15.6%

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

      7. distribute-rgt-neg-in 15.6%

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

      8. *-commutative 15.6%

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

      9. distribute-rgt-neg-in 15.6%

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

      10. remove-double-neg 15.6%

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

      11. metadata-eval 15.6%

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

   10. Applied egg-rr 15.6%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.4% accurate, 22.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 97.3%

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

      1. fma-neg 97.3%

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

      2. /-rgt-identity 97.3%

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

      3. fma-neg 97.3%

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

      4. /-rgt-identity 97.3%

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

      5. distribute-rgt-neg-in 97.3%

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

      6. sub-neg 97.3%

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

      7. metadata-eval 97.3%

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

      8. distribute-rgt-neg-in 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in x around 0 73.8%

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

   6. Taylor expanded in eps around inf 49.6%

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

      1. associate-*r* 49.6%

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

      2. mul-1-neg 49.6%

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

   8. Simplified 49.6%

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

      1. frac-2neg 49.6%

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

      2. div-inv 49.6%

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

      3. *-commutative 49.6%

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

      4. distribute-rgt-neg-in 49.6%

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

      5. remove-double-neg 49.6%

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

      6. metadata-eval 49.6%

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

      7. metadata-eval 49.6%

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

   10. Applied egg-rr 49.6%

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

   if -1 < x

   1. Initial program 68.7%

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

      1. fma-neg 68.7%

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

      2. /-rgt-identity 68.7%

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

      3. fma-neg 68.7%

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

      4. /-rgt-identity 68.7%

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

      5. distribute-rgt-neg-in 68.7%

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

      6. sub-neg 68.7%

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

      7. metadata-eval 68.7%

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

      8. distribute-rgt-neg-in 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in x around 0 50.0%

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

4. Final simplification 49.9%

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

Alternative 14: 9.7% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.5)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.5;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.5d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.5;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.5

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.5
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.5;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.5

Derivation

1. Initial program 72.8%

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

   1. fma-neg 72.8%

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

   2. /-rgt-identity 72.8%

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

   3. fma-neg 72.8%

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

   4. /-rgt-identity 72.8%

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

   5. distribute-rgt-neg-in 72.8%

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

   6. sub-neg 72.8%

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

   7. metadata-eval 72.8%

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

   8. distribute-rgt-neg-in 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in x around 0 42.0%

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

6. Taylor expanded in x around 0 28.6%

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

7. Taylor expanded in eps around 0 17.1%

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

8. Taylor expanded in eps around 0 9.7%

   \[\leadsto \frac{\color{blue}{1}}{2} \]

9. Final simplification 9.7%

   \[\leadsto 0.5 \]
10. Add Preprocessing

Alternative 15: 43.5% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. fma-neg 72.8%

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

   2. /-rgt-identity 72.8%

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

   3. fma-neg 72.8%

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

   4. /-rgt-identity 72.8%

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

   5. distribute-rgt-neg-in 72.8%

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

   6. sub-neg 72.8%

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

   7. metadata-eval 72.8%

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

   8. distribute-rgt-neg-in 72.8%

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

3. Simplified 72.8%

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

5. Taylor expanded in x around 0 43.2%

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

6. Final simplification 43.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))