NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.7%

Time: 16.7s

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: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x + 1}{e^{x}}\\ \mathbf{if}\;eps\_m \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{\left(-x\right) \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ x 1.0) (exp x))))
   (if (<= eps_m 2e-25)
     (/ (+ t_0 t_0) 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 t_0 = (x + 1.0) / exp(x);
	double tmp;
	if (eps_m <= 2e-25) {
		tmp = (t_0 + t_0) / 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) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) / exp(x)
    if (eps_m <= 2d-25) then
        tmp = (t_0 + t_0) / 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 t_0 = (x + 1.0) / Math.exp(x);
	double tmp;
	if (eps_m <= 2e-25) {
		tmp = (t_0 + t_0) / 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):
	t_0 = (x + 1.0) / math.exp(x)
	tmp = 0
	if eps_m <= 2e-25:
		tmp = (t_0 + t_0) / 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)
	t_0 = Float64(Float64(x + 1.0) / exp(x))
	tmp = 0.0
	if (eps_m <= 2e-25)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(Float64(-x) * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x + 1.0) / exp(x);
	tmp = 0.0;
	if (eps_m <= 2e-25)
		tmp = (t_0 + t_0) / 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_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 2e-25], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[((-x) * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.00000000000000008e-25

   1. Initial program 65.6%

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

      1. fma-neg 65.6%

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

      2. /-rgt-identity 65.6%

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

      3. fma-neg 65.6%

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

      4. /-rgt-identity 65.6%

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

      5. distribute-rgt-neg-in 65.6%

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

      6. sub-neg 65.6%

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

      7. metadata-eval 65.6%

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

      8. distribute-rgt-neg-in 65.6%

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

   3. Simplified 65.6%

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

   5. Taylor expanded in eps around 0 69.5%

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

   7. Simplified 69.5%

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

      1. exp-neg 69.5%

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

      2. un-div-inv 69.5%

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

   9. Applied egg-rr 69.5%

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

       1. exp-neg 69.5%

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

       2. un-div-inv 69.5%

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

   11. Applied egg-rr 69.5%

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

   if 2.00000000000000008e-25 < eps

   1. Initial program 99.9%

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

      1. fma-neg 99.9%

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

      2. /-rgt-identity 99.9%

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

      3. fma-neg 99.9%

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

      4. /-rgt-identity 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-rgt-neg-in 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in eps around inf 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. neg-mul-1 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around inf 99.9%

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

       1. mul-1-neg 99.9%

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

       2. distribute-rgt-neg-in 99.9%

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

       3. associate-*r* 99.9%

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

       4. mul-1-neg 99.9%

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

   13. Simplified 99.9%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \frac{x + 1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\left(-x\right) \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))
   (if (<= x -2.6e-285)
     (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
     (if (<= x 2400.0)
       t_0
       (if (<= x 1.65e+184)
         (/ (/ (+ (exp (- x)) (/ -1.0 (exp x))) eps_m) 2.0)
         (if (<= x 3.2e+208)
           t_0
           (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	double tmp;
	if (x <= -2.6e-285) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2400.0) {
		tmp = t_0;
	} else if (x <= 1.65e+184) {
		tmp = ((exp(-x) + (-1.0 / exp(x))) / eps_m) / 2.0;
	} else if (x <= 3.2e+208) {
		tmp = t_0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    if (x <= (-2.6d-285)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 2400.0d0) then
        tmp = t_0
    else if (x <= 1.65d+184) then
        tmp = ((exp(-x) + ((-1.0d0) / exp(x))) / eps_m) / 2.0d0
    else if (x <= 3.2d+208) then
        tmp = t_0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	double tmp;
	if (x <= -2.6e-285) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 2400.0) {
		tmp = t_0;
	} else if (x <= 1.65e+184) {
		tmp = ((Math.exp(-x) + (-1.0 / Math.exp(x))) / eps_m) / 2.0;
	} else if (x <= 3.2e+208) {
		tmp = t_0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	tmp = 0
	if x <= -2.6e-285:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 2400.0:
		tmp = t_0
	elif x <= 1.65e+184:
		tmp = ((math.exp(-x) + (-1.0 / math.exp(x))) / eps_m) / 2.0
	elif x <= 3.2e+208:
		tmp = t_0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0)
	tmp = 0.0
	if (x <= -2.6e-285)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 2400.0)
		tmp = t_0;
	elseif (x <= 1.65e+184)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) + Float64(-1.0 / exp(x))) / eps_m) / 2.0);
	elseif (x <= 3.2e+208)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	tmp = 0.0;
	if (x <= -2.6e-285)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 2400.0)
		tmp = t_0;
	elseif (x <= 1.65e+184)
		tmp = ((exp(-x) + (-1.0 / exp(x))) / eps_m) / 2.0;
	elseif (x <= 3.2e+208)
		tmp = t_0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.6e-285], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2400.0], t$95$0, If[LessEqual[x, 1.65e+184], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] + N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.2e+208], 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6000000000000002e-285

   1. Initial program 78.6%

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

      1. fma-neg 78.6%

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

      2. /-rgt-identity 78.6%

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

      3. fma-neg 78.6%

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

      4. /-rgt-identity 78.6%

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

      5. distribute-rgt-neg-in 78.6%

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

      6. sub-neg 78.6%

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

      7. metadata-eval 78.6%

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

      8. distribute-rgt-neg-in 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in x around 0 47.2%

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

   6. Taylor expanded in eps around inf 67.9%

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

   if -2.6000000000000002e-285 < x < 2400 or 1.6499999999999999e184 < x < 3.2000000000000001e208

   1. Initial program 52.4%

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

      1. fma-neg 52.4%

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

      2. /-rgt-identity 52.4%

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

      3. fma-neg 52.4%

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

      4. /-rgt-identity 52.4%

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

      5. distribute-rgt-neg-in 52.4%

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

      6. sub-neg 52.4%

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

      7. metadata-eval 52.4%

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

      8. distribute-rgt-neg-in 52.4%

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

   3. Simplified 52.4%

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

   5. Taylor expanded in eps around inf 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in eps around inf 96.2%

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

      1. associate-*r* 96.2%

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

      2. neg-mul-1 96.2%

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

   10. Simplified 96.2%

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

   11. Taylor expanded in x around inf 96.2%

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

       1. mul-1-neg 96.2%

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

       2. distribute-rgt-neg-in 96.2%

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

       3. associate-*r* 96.2%

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

       4. mul-1-neg 96.2%

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

   13. Simplified 96.2%

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

   14. Taylor expanded in eps around 0 83.2%

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

   if 2400 < x < 1.6499999999999999e184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 59.5%

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

   if 3.2000000000000001e208 < 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 12.8%

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

   6. Taylor expanded in x around 0 72.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{e^{-x} + \frac{-1}{e^{x}}}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+208}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 3.8:\\ \;\;\;\;\frac{t\_0 + e^{\left(-x\right) \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 3.8)
     (/ (+ 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 <= 3.8) {
		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 <= 3.8d0) 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 <= 3.8) {
		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 <= 3.8:
		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 <= 3.8)
		tmp = Float64(Float64(t_0 + exp(Float64(Float64(-x) * 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 <= 3.8)
		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, 3.8], 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 < 3.7999999999999998

   1. Initial program 65.5%

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

      1. fma-neg 65.6%

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

      2. /-rgt-identity 65.6%

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

      3. fma-neg 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. distribute-rgt-neg-in 65.5%

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

      6. sub-neg 65.5%

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

      7. metadata-eval 65.5%

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

      8. distribute-rgt-neg-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in eps around inf 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in eps around inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

   10. Simplified 98.8%

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

   11. Taylor expanded in x around inf 98.8%

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

       1. mul-1-neg 98.8%

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

       2. distribute-rgt-neg-in 98.8%

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

       3. associate-*r* 98.8%

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

       4. mul-1-neg 98.8%

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

   13. Simplified 98.8%

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

   if 3.7999999999999998 < x

   1. Initial program 98.6%

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

      1. fma-neg 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. fma-neg 98.6%

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

      4. /-rgt-identity 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

      6. sub-neg 98.6%

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

      7. metadata-eval 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in eps around inf 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in eps around 0 76.3%

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

      1. neg-mul-1 76.3%

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

   10. Simplified 76.3%

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

4. Final simplification 92.6%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-284)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (- x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-284) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + 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) :: tmp
    if (x <= (-5d-284)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + 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 tmp;
	if (x <= -5e-284) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-284:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp(-x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-284)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-284)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-284], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999973e-284

   1. Initial program 78.6%

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

      1. fma-neg 78.6%

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

      2. /-rgt-identity 78.6%

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

      3. fma-neg 78.6%

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

      4. /-rgt-identity 78.6%

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

      5. distribute-rgt-neg-in 78.6%

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

      6. sub-neg 78.6%

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

      7. metadata-eval 78.6%

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

      8. distribute-rgt-neg-in 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in x around 0 47.2%

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

   6. Taylor expanded in eps around inf 67.9%

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

   if -4.99999999999999973e-284 < x

   1. Initial program 71.8%

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

      1. fma-neg 71.8%

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

      2. /-rgt-identity 71.8%

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

      3. fma-neg 71.8%

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

      4. /-rgt-identity 71.8%

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

      5. distribute-rgt-neg-in 71.8%

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

      6. sub-neg 71.8%

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

      7. metadata-eval 71.8%

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

      8. distribute-rgt-neg-in 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in eps around inf 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in eps around 0 81.6%

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

      1. neg-mul-1 81.6%

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

   10. Simplified 81.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% 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 74.6%

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

   1. fma-neg 74.6%

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

   2. /-rgt-identity 74.6%

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

   3. fma-neg 74.6%

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

   4. /-rgt-identity 74.6%

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

   5. distribute-rgt-neg-in 74.6%

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

   6. sub-neg 74.6%

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

   7. metadata-eval 74.6%

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

   8. distribute-rgt-neg-in 74.6%

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

3. Simplified 74.6%

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

5. Taylor expanded in eps around inf 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 6: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+209}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5e-280)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 3.8e+209)
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-280) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 3.8e+209) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-5d-280)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 3.8d+209) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -5e-280) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 3.8e+209) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5e-280:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 3.8e+209:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5e-280)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 3.8e+209)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -5e-280)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 3.8e+209)
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -5e-280], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e+209], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000028e-280

   1. Initial program 78.6%

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

      1. fma-neg 78.6%

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

      2. /-rgt-identity 78.6%

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

      3. fma-neg 78.6%

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

      4. /-rgt-identity 78.6%

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

      5. distribute-rgt-neg-in 78.6%

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

      6. sub-neg 78.6%

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

      7. metadata-eval 78.6%

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

      8. distribute-rgt-neg-in 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in eps around inf 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in eps around inf 99.3%

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

      1. associate-*r* 99.3%

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

      2. neg-mul-1 99.3%

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

   10. Simplified 99.3%

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

   11. Taylor expanded in eps around 0 84.1%

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

       1. mul-1-neg 84.1%

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

   13. Simplified 84.1%

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

   if -5.00000000000000028e-280 < x < 3.79999999999999984e209

   1. Initial program 65.5%

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

      1. fma-neg 65.5%

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

      2. /-rgt-identity 65.5%

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

      3. fma-neg 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. distribute-rgt-neg-in 65.5%

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

      6. sub-neg 65.5%

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

      7. metadata-eval 65.5%

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

      8. distribute-rgt-neg-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in eps around inf 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in eps around inf 87.1%

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

      1. associate-*r* 87.1%

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

      2. neg-mul-1 87.1%

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

   10. Simplified 87.1%

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

   11. Taylor expanded in x around inf 87.1%

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

       1. mul-1-neg 87.1%

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

       2. distribute-rgt-neg-in 87.1%

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

       3. associate-*r* 87.1%

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

       4. mul-1-neg 87.1%

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

   13. Simplified 87.1%

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

   14. Taylor expanded in eps around 0 63.6%

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

   if 3.79999999999999984e209 < 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 12.8%

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

   6. Taylor expanded in x around 0 72.3%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-280}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+209}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+211}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-284)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 1.8e+211)
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-284) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.8e+211) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-284)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 1.8d+211) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-284) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.8e+211) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-284:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 1.8e+211:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-284)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.8e+211)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-284)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.8e+211)
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-284], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8e+211], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000004e-284

   1. Initial program 78.6%

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

      1. fma-neg 78.6%

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

      2. /-rgt-identity 78.6%

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

      3. fma-neg 78.6%

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

      4. /-rgt-identity 78.6%

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

      5. distribute-rgt-neg-in 78.6%

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

      6. sub-neg 78.6%

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

      7. metadata-eval 78.6%

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

      8. distribute-rgt-neg-in 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in x around 0 47.2%

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

   6. Taylor expanded in eps around inf 67.9%

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

   if -1.00000000000000004e-284 < x < 1.80000000000000001e211

   1. Initial program 65.5%

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

      1. fma-neg 65.5%

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

      2. /-rgt-identity 65.5%

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

      3. fma-neg 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. distribute-rgt-neg-in 65.5%

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

      6. sub-neg 65.5%

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

      7. metadata-eval 65.5%

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

      8. distribute-rgt-neg-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in eps around inf 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in eps around inf 87.1%

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

      1. associate-*r* 87.1%

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

      2. neg-mul-1 87.1%

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

   10. Simplified 87.1%

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

   11. Taylor expanded in x around inf 87.1%

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

       1. mul-1-neg 87.1%

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

       2. distribute-rgt-neg-in 87.1%

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

       3. associate-*r* 87.1%

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

       4. mul-1-neg 87.1%

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

   13. Simplified 87.1%

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

   14. Taylor expanded in eps around 0 63.6%

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

   if 1.80000000000000001e211 < 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 12.8%

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

   6. Taylor expanded in x around 0 72.3%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+211}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 360.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 360.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 360.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. /-rgt-identity 65.2%

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

      3. fma-neg 65.2%

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

      4. /-rgt-identity 65.2%

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

      5. distribute-rgt-neg-in 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in eps around inf 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in eps around inf 98.3%

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

      1. associate-*r* 98.3%

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

      2. neg-mul-1 98.3%

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

   10. Simplified 98.3%

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

   11. Taylor expanded in eps around 0 80.1%

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

       1. mul-1-neg 80.1%

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

   13. Simplified 80.1%

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

   if 360 < 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 25.1%

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

   6. Taylor expanded in x around 0 56.5%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{\left(1 - x \cdot eps\_m\right) + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 200.0)
   (/ (+ (- 1.0 (* x eps_m)) (+ x (- 1.0 x))) 2.0)
   (/ (+ (/ 1.0 eps_m) (- 1.0 (/ 1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 200.0) {
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = ((1.0 / eps_m) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 200.0d0) then
        tmp = ((1.0d0 - (x * eps_m)) + (x + (1.0d0 - x))) / 2.0d0
    else
        tmp = ((1.0d0 / eps_m) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 200.0) {
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = ((1.0 / eps_m) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 200.0:
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0
	else:
		tmp = ((1.0 / eps_m) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 200.0)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * eps_m)) + Float64(x + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 200.0)
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0;
	else
		tmp = ((1.0 / eps_m) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 200.0], N[(N[(N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 200

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. /-rgt-identity 65.2%

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

      3. fma-neg 65.2%

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

      4. /-rgt-identity 65.2%

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

      5. distribute-rgt-neg-in 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in x around 0 27.7%

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

   7. Taylor expanded in eps around inf 61.2%

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

      1. sub-neg 61.2%

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

      2. mul-1-neg 61.2%

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

      3. unsub-neg 61.2%

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

      4. distribute-lft-out 61.2%

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

      5. distribute-lft-out 61.2%

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

      6. mul-1-neg 61.2%

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

      7. +-commutative 61.2%

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

      8. mul-1-neg 61.2%

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

      9. metadata-eval 61.2%

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

      10. distribute-neg-in 61.2%

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

      11. remove-double-neg 61.2%

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

      12. distribute-neg-in 61.2%

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

      13. mul-1-neg 61.2%

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

      14. associate-*r* 61.2%

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

      15. metadata-eval 61.2%

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

      16. *-lft-identity 61.2%

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

      17. distribute-neg-in 61.2%

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

      18. mul-1-neg 61.2%

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

      19. metadata-eval 61.2%

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

      20. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 200 < 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 25.1%

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

   6. Taylor expanded in x around 0 56.5%

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

   7. Taylor expanded in eps around 0 56.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{\left(1 - x \cdot \varepsilon\right) + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 190:\\ \;\;\;\;\frac{\left(1 - x \cdot eps\_m\right) + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 190.0)
   (/ (+ (- 1.0 (* x eps_m)) (+ x (- 1.0 x))) 2.0)
   (/ (+ (+ 1.0 (/ 1.0 eps_m)) (- 1.0 (/ 1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 190.0) {
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 190.0d0) then
        tmp = ((1.0d0 - (x * eps_m)) + (x + (1.0d0 - x))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 190.0) {
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 190.0:
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 190.0)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * eps_m)) + Float64(x + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 190.0)
		tmp = ((1.0 - (x * eps_m)) + (x + (1.0 - x))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 190.0], N[(N[(N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 190

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. /-rgt-identity 65.2%

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

      3. fma-neg 65.2%

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

      4. /-rgt-identity 65.2%

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

      5. distribute-rgt-neg-in 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around 0 43.0%

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

   6. Taylor expanded in x around 0 27.7%

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

   7. Taylor expanded in eps around inf 61.2%

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

      1. sub-neg 61.2%

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

      2. mul-1-neg 61.2%

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

      3. unsub-neg 61.2%

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

      4. distribute-lft-out 61.2%

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

      5. distribute-lft-out 61.2%

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

      6. mul-1-neg 61.2%

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

      7. +-commutative 61.2%

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

      8. mul-1-neg 61.2%

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

      9. metadata-eval 61.2%

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

      10. distribute-neg-in 61.2%

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

      11. remove-double-neg 61.2%

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

      12. distribute-neg-in 61.2%

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

      13. mul-1-neg 61.2%

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

      14. associate-*r* 61.2%

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

      15. metadata-eval 61.2%

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

      16. *-lft-identity 61.2%

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

      17. distribute-neg-in 61.2%

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

      18. mul-1-neg 61.2%

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

      19. metadata-eval 61.2%

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

      20. +-commutative 61.2%

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

   9. Simplified 61.2%

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

   if 190 < 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 25.1%

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

   6. Taylor expanded in x around 0 56.5%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 190:\\ \;\;\;\;\frac{\left(1 - x \cdot \varepsilon\right) + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.6% accurate, 14.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 65.5%

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

      1. fma-neg 65.6%

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

      2. /-rgt-identity 65.6%

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

      3. fma-neg 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. distribute-rgt-neg-in 65.5%

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

      6. sub-neg 65.5%

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

      7. metadata-eval 65.5%

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

      8. distribute-rgt-neg-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around 0 43.2%

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

   6. Taylor expanded in eps around inf 76.5%

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

   7. Taylor expanded in x around 0 61.1%

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

   if 2 < x

   1. Initial program 98.6%

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

      1. fma-neg 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. fma-neg 98.6%

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

      4. /-rgt-identity 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

      6. sub-neg 98.6%

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

      7. metadata-eval 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in x around 0 24.7%

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

   6. Taylor expanded in x around 0 55.7%

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

   7. Taylor expanded in eps around 0 55.7%

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

4. Final simplification 59.6%

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

Alternative 12: 58.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\left(-x\right) \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 185:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (/ (* (- x) eps_m) 2.0)
   (if (<= x 185.0) 1.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-x * eps_m) / 2.0;
	} else if (x <= 185.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (-x * eps_m) / 2.0d0
    else if (x <= 185.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (-x * eps_m) / 2.0;
	} else if (x <= 185.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (-x * eps_m) / 2.0
	elif x <= 185.0:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(Float64(-x) * eps_m) / 2.0);
	elseif (x <= 185.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (-x * eps_m) / 2.0;
	elseif (x <= 185.0)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[((-x) * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 185.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   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 37.7%

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

   6. Taylor expanded in x around 0 15.7%

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

   7. Taylor expanded in eps around inf 15.7%

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

      1. mul-1-neg 15.7%

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

      2. *-commutative 15.7%

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

      3. distribute-rgt-neg-in 15.7%

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

   9. Simplified 15.7%

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

   if -1 < x < 185

   1. Initial program 55.1%

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

      1. fma-neg 55.1%

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

      2. /-rgt-identity 55.1%

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

      3. fma-neg 55.1%

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

      4. /-rgt-identity 55.1%

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

      5. distribute-rgt-neg-in 55.1%

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

      6. sub-neg 55.1%

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

      7. metadata-eval 55.1%

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

      8. distribute-rgt-neg-in 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in x around 0 75.5%

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

   if 185 < 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 22.2%

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

   6. Taylor expanded in eps around inf 9.9%

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

      1. *-commutative 9.9%

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

   8. Simplified 9.9%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\left(-x\right) \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 185:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 16.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 65.5%

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

      1. fma-neg 65.6%

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

      2. /-rgt-identity 65.6%

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

      3. fma-neg 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. distribute-rgt-neg-in 65.5%

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

      6. sub-neg 65.5%

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

      7. metadata-eval 65.5%

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

      8. distribute-rgt-neg-in 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around 0 43.2%

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

   6. Taylor expanded in eps around inf 76.5%

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

   7. Taylor expanded in x around 0 61.1%

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

   if 2 < x

   1. Initial program 98.6%

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

      1. fma-neg 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. fma-neg 98.6%

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

      4. /-rgt-identity 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

      6. sub-neg 98.6%

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

      7. metadata-eval 98.6%

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

      8. distribute-rgt-neg-in 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in x around 0 21.9%

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

   6. Taylor expanded in eps around inf 9.8%

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

      1. *-commutative 9.8%

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

   8. Simplified 9.8%

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

4. Final simplification 47.1%

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

Alternative 14: 50.9% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 185:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 185.0) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 185.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 185.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 185.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 185.0:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 185.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 185.0)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 185.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 185

   1. Initial program 65.2%

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

      1. fma-neg 65.2%

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

      2. /-rgt-identity 65.2%

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

      3. fma-neg 65.2%

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

      4. /-rgt-identity 65.2%

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

      5. distribute-rgt-neg-in 65.2%

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

      6. sub-neg 65.2%

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

      7. metadata-eval 65.2%

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

      8. distribute-rgt-neg-in 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around 0 59.3%

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

   if 185 < 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 22.2%

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

   6. Taylor expanded in eps around inf 9.9%

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

      1. *-commutative 9.9%

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

   8. Simplified 9.9%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 185:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.6% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. fma-neg 74.6%

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

   2. /-rgt-identity 74.6%

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

   3. fma-neg 74.6%

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

   4. /-rgt-identity 74.6%

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

   5. distribute-rgt-neg-in 74.6%

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

   6. sub-neg 74.6%

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

   7. metadata-eval 74.6%

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

   8. distribute-rgt-neg-in 74.6%

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

3. Simplified 74.6%

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

5. Taylor expanded in x around 0 44.1%

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

6. Final simplification 44.1%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))