NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.5%

Time: 17.6s

Alternatives: 17

Speedup: 1.1×

• 38.0% 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: 73.4% 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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 5.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps_m - 1\right)} + e^{x \cdot \left(-eps_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.2e-74)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)
   (/ (+ (exp (* x (- eps_m 1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.2e-74) {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m - 1.0))) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5.2d-74) then
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    else
        tmp = (exp((x * (eps_m - 1.0d0))) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.2e-74) {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m - 1.0))) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5.2e-74:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m - 1.0))) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5.2e-74)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m - 1.0))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5.2e-74)
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	else
		tmp = (exp((x * (eps_m - 1.0))) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5.2e-74], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.2000000000000002e-74

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in eps around 0 68.5%

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

   6. Simplified 68.5%

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

      1. rec-exp 68.5%

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

      2. un-div-inv 68.5%

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

   8. Applied egg-rr 68.5%

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

   if 5.2000000000000002e-74 < eps

   1. Initial program 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. mul-1-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. remove-double-neg 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 77.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-282} \lor \neg \left(x \leq 3.4 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(eps_m, eps_m, -1\right)}{\frac{1 + eps_m}{1 + \frac{1}{eps_m}}} + \left(1 - \frac{1}{eps_m}\right) \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 6.5e-282) (not (<= x 3.4e+169)))
   (/ (+ (exp (- x)) (exp (* x (- eps_m)))) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       (/ (fma eps_m eps_m -1.0) (/ (+ 1.0 eps_m) (+ 1.0 (/ 1.0 eps_m))))
       (* (- 1.0 (/ 1.0 eps_m)) (- -1.0 eps_m)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 6.5e-282) || !(x <= 3.4e+169)) {
		tmp = (exp(-x) + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((fma(eps_m, eps_m, -1.0) / ((1.0 + eps_m) / (1.0 + (1.0 / eps_m)))) + ((1.0 - (1.0 / eps_m)) * (-1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 6.5e-282) || !(x <= 3.4e+169))
		tmp = Float64(Float64(exp(Float64(-x)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(fma(eps_m, eps_m, -1.0) / Float64(Float64(1.0 + eps_m) / Float64(1.0 + Float64(1.0 / eps_m)))) + Float64(Float64(1.0 - Float64(1.0 / eps_m)) * Float64(-1.0 - eps_m))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 6.5e-282], N[Not[LessEqual[x, 3.4e+169]], $MachinePrecision]], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(N[(1.0 + eps$95$m), $MachinePrecision] / N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000012e-282 or 3.40000000000000028e169 < x

   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

   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. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in eps around inf 93.6%

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

      1. *-commutative 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in x around inf 93.6%

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

      1. sub-neg 93.6%

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

      2. neg-mul-1 93.6%

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

      3. distribute-lft-neg-in 93.6%

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

      4. mul-1-neg 93.6%

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

      5. mul-1-neg 93.6%

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

      6. *-commutative 93.6%

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

      7. remove-double-neg 93.6%

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

      8. distribute-rgt-neg-in 93.6%

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

   10. Simplified 93.6%

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

   11. Taylor expanded in eps around 0 86.4%

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

       1. mul-1-neg 86.4%

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

   13. Simplified 86.4%

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

   if 6.50000000000000012e-282 < x < 3.40000000000000028e169

   1. Initial program 70.1%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in x around 0 46.6%

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

      1. flip-- 56.3%

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

      2. +-commutative 56.3%

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

      3. associate-*r/ 56.3%

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

      4. metadata-eval 56.3%

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

      5. fma-neg 56.3%

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

      6. metadata-eval 56.3%

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

   6. Applied egg-rr 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-/l* 56.3%

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

      3. +-commutative 56.3%

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

   8. Simplified 56.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-282} \lor \neg \left(x \leq 3.4 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\frac{1 + \varepsilon}{1 + \frac{1}{\varepsilon}}} + \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 3: 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 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in eps around inf 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 92.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-eps_m\right)}\\ \mathbf{if}\;eps_m \leq 0.4:\\ \;\;\;\;\frac{e^{-x} + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + e^{x \cdot eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- eps_m)))))
   (if (<= eps_m 0.4)
     (/ (+ (exp (- x)) t_0) 2.0)
     (/ (+ t_0 (exp (* x eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * -eps_m));
	double tmp;
	if (eps_m <= 0.4) {
		tmp = (exp(-x) + t_0) / 2.0;
	} else {
		tmp = (t_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 = exp((x * -eps_m))
    if (eps_m <= 0.4d0) then
        tmp = (exp(-x) + t_0) / 2.0d0
    else
        tmp = (t_0 + 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 = Math.exp((x * -eps_m));
	double tmp;
	if (eps_m <= 0.4) {
		tmp = (Math.exp(-x) + t_0) / 2.0;
	} else {
		tmp = (t_0 + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * -eps_m))
	tmp = 0
	if eps_m <= 0.4:
		tmp = (math.exp(-x) + t_0) / 2.0
	else:
		tmp = (t_0 + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-eps_m)))
	tmp = 0.0
	if (eps_m <= 0.4)
		tmp = Float64(Float64(exp(Float64(-x)) + t_0) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * -eps_m));
	tmp = 0.0;
	if (eps_m <= 0.4)
		tmp = (exp(-x) + t_0) / 2.0;
	else
		tmp = (t_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[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.4], N[(N[(N[Exp[(-x)], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.40000000000000002

   1. Initial program 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in eps around inf 99.0%

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

   5. Taylor expanded in eps around inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in x around inf 85.6%

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

      1. sub-neg 85.6%

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

      2. neg-mul-1 85.6%

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

      3. distribute-lft-neg-in 85.6%

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

      4. mul-1-neg 85.6%

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

      5. mul-1-neg 85.6%

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

      6. *-commutative 85.6%

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

      7. remove-double-neg 85.6%

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

      8. distribute-rgt-neg-in 85.6%

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

   10. Simplified 85.6%

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

   11. Taylor expanded in eps around 0 81.7%

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

       1. mul-1-neg 81.7%

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

   13. Simplified 81.7%

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

   if 0.40000000000000002 < eps

   1. Initial program 100.0%

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

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. mul-1-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. remove-double-neg 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.4:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-180}:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps_m\right)} + e^{x \cdot eps_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps_m - 1\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1e-180)
   (/ (+ (exp (* x (- eps_m))) (exp (* x 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 <= 1e-180) {
		tmp = (exp((x * -eps_m)) + exp((x * 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 <= 1d-180) then
        tmp = (exp((x * -eps_m)) + exp((x * 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 <= 1e-180) {
		tmp = (Math.exp((x * -eps_m)) + Math.exp((x * 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 <= 1e-180:
		tmp = (math.exp((x * -eps_m)) + math.exp((x * 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 <= 1e-180)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + exp(Float64(x * 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 <= 1e-180)
		tmp = (exp((x * -eps_m)) + exp((x * 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, 1e-180], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $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 < 1e-180

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around inf 98.9%

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

      1. sub-neg 98.9%

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

      2. neg-mul-1 98.9%

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

      3. distribute-lft-neg-in 98.9%

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

      4. mul-1-neg 98.9%

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

      5. mul-1-neg 98.9%

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

      6. *-commutative 98.9%

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

      7. remove-double-neg 98.9%

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

      8. distribute-rgt-neg-in 98.9%

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

   10. Simplified 98.9%

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

   11. Taylor expanded in eps around inf 98.9%

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

       1. *-commutative 98.9%

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

   13. Simplified 98.9%

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

   if 1e-180 < x

   1. Initial program 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 85.1%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-180}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon - 1\right)} + e^{-x}}{2}\\ \end{array} \]

Alternative 6: 92.1% accurate, 1.1× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (- eps_m 1.0))) (exp (* x (- 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 * -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 * -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 * -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 * -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(-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 * -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 * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in eps around inf 99.3%

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

5. Taylor expanded in eps around inf 90.2%

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

   1. *-commutative 90.2%

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

7. Simplified 90.2%

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

8. Taylor expanded in x around inf 90.2%

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

   1. sub-neg 90.2%

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

   2. neg-mul-1 90.2%

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

   3. distribute-lft-neg-in 90.2%

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

   4. mul-1-neg 90.2%

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

   5. mul-1-neg 90.2%

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

   6. *-commutative 90.2%

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

   7. remove-double-neg 90.2%

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

   8. distribute-rgt-neg-in 90.2%

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

10. Simplified 90.2%

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

11. Final simplification 90.2%

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

Alternative 7: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 - \frac{1}{eps_m}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-42}:\\ \;\;\;\;\frac{2 + x \cdot \left(eps_m + \frac{-1 + {eps_m}^{2}}{\frac{1 - eps_m}{t_0}}\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-282}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(eps_m, eps_m, -1\right)}{\frac{1 + eps_m}{1 + \frac{1}{eps_m}}} + t_0 \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 1.0 eps_m))))
   (if (<= x -7.5e+34)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x -8e-42)
       (/
        (+
         2.0
         (* x (+ eps_m (/ (+ -1.0 (pow eps_m 2.0)) (/ (- 1.0 eps_m) t_0)))))
        2.0)
       (if (<= x 6.2e-282)
         1.0
         (/
          (+
           2.0
           (*
            x
            (+
             (/ (fma eps_m eps_m -1.0) (/ (+ 1.0 eps_m) (+ 1.0 (/ 1.0 eps_m))))
             (* t_0 (- -1.0 eps_m)))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 - (1.0 / eps_m);
	double tmp;
	if (x <= -7.5e+34) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -8e-42) {
		tmp = (2.0 + (x * (eps_m + ((-1.0 + pow(eps_m, 2.0)) / ((1.0 - eps_m) / t_0))))) / 2.0;
	} else if (x <= 6.2e-282) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (x * ((fma(eps_m, eps_m, -1.0) / ((1.0 + eps_m) / (1.0 + (1.0 / eps_m)))) + (t_0 * (-1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 - Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -7.5e+34)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -8e-42)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps_m + Float64(Float64(-1.0 + (eps_m ^ 2.0)) / Float64(Float64(1.0 - eps_m) / t_0))))) / 2.0);
	elseif (x <= 6.2e-282)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(fma(eps_m, eps_m, -1.0) / Float64(Float64(1.0 + eps_m) / Float64(1.0 + Float64(1.0 / eps_m)))) + Float64(t_0 * Float64(-1.0 - eps_m))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+34], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -8e-42], N[(N[(2.0 + N[(x * N[(eps$95$m + N[(N[(-1.0 + N[Power[eps$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-282], 1.0, N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(N[(1.0 + eps$95$m), $MachinePrecision] / N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.49999999999999976e34

   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

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   if -7.49999999999999976e34 < x < -8.0000000000000003e-42

   1. Initial program 70.9%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in x around 0 24.1%

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

   5. Taylor expanded in eps around inf 4.3%

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

      1. *-commutative 4.3%

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

      2. flip-+ 47.3%

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

      3. associate-*r/ 47.3%

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

      4. metadata-eval 47.3%

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

      5. pow2 47.3%

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

   7. Applied egg-rr 47.3%

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

      1. *-commutative 47.3%

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

      2. associate-/l* 47.3%

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

   9. Simplified 47.3%

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

   if -8.0000000000000003e-42 < x < 6.20000000000000027e-282

   1. Initial program 51.6%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in x around 0 86.8%

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

   if 6.20000000000000027e-282 < x

   1. Initial program 75.8%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in x around 0 38.3%

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

      1. flip-- 48.4%

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

      2. +-commutative 48.4%

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

      3. associate-*r/ 48.4%

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

      4. metadata-eval 48.4%

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

      5. fma-neg 48.4%

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

      6. metadata-eval 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-/l* 48.4%

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

      3. +-commutative 48.4%

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

   8. Simplified 48.4%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-42}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1 + {\varepsilon}^{2}}{\frac{1 - \varepsilon}{1 - \frac{1}{\varepsilon}}}\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-282}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\frac{1 + \varepsilon}{1 + \frac{1}{\varepsilon}}} + \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 8: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps_m}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-234}:\\ \;\;\;\;\frac{2 + x \cdot \left(t_0 \cdot \left(eps_m - 1\right) + \left(1 - {eps_m}^{2}\right) \cdot \frac{-1 + \frac{1}{eps_m}}{1 - eps_m}\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-281}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(eps_m, eps_m, -1\right)}{\frac{1 + eps_m}{t_0}} + \left(1 - \frac{1}{eps_m}\right) \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -7.5e+34)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x -3.4e-234)
       (/
        (+
         2.0
         (*
          x
          (+
           (* t_0 (- eps_m 1.0))
           (*
            (- 1.0 (pow eps_m 2.0))
            (/ (+ -1.0 (/ 1.0 eps_m)) (- 1.0 eps_m))))))
        2.0)
       (if (<= x 1.8e-281)
         1.0
         (/
          (+
           2.0
           (*
            x
            (+
             (/ (fma eps_m eps_m -1.0) (/ (+ 1.0 eps_m) t_0))
             (* (- 1.0 (/ 1.0 eps_m)) (- -1.0 eps_m)))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -7.5e+34) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -3.4e-234) {
		tmp = (2.0 + (x * ((t_0 * (eps_m - 1.0)) + ((1.0 - pow(eps_m, 2.0)) * ((-1.0 + (1.0 / eps_m)) / (1.0 - eps_m)))))) / 2.0;
	} else if (x <= 1.8e-281) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (x * ((fma(eps_m, eps_m, -1.0) / ((1.0 + eps_m) / t_0)) + ((1.0 - (1.0 / eps_m)) * (-1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -7.5e+34)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -3.4e-234)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(t_0 * Float64(eps_m - 1.0)) + Float64(Float64(1.0 - (eps_m ^ 2.0)) * Float64(Float64(-1.0 + Float64(1.0 / eps_m)) / Float64(1.0 - eps_m)))))) / 2.0);
	elseif (x <= 1.8e-281)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(fma(eps_m, eps_m, -1.0) / Float64(Float64(1.0 + eps_m) / t_0)) + Float64(Float64(1.0 - Float64(1.0 / eps_m)) * Float64(-1.0 - eps_m))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+34], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -3.4e-234], N[(N[(2.0 + N[(x * N[(N[(t$95$0 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[Power[eps$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8e-281], 1.0, N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(N[(1.0 + eps$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.49999999999999976e34

   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

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   if -7.49999999999999976e34 < x < -3.39999999999999986e-234

   1. Initial program 59.0%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. *-commutative 37.3%

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

      2. flip-+ 60.0%

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

      3. associate-*r/ 60.0%

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

      4. metadata-eval 60.0%

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

      5. pow2 60.0%

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

   6. Applied egg-rr 82.2%

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

      1. associate-/l* 82.2%

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

      2. associate-/r/ 82.2%

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

   8. Simplified 82.2%

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

   if -3.39999999999999986e-234 < x < 1.80000000000000003e-281

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.80000000000000003e-281 < x

   1. Initial program 75.8%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in x around 0 38.3%

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

      1. flip-- 48.4%

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

      2. +-commutative 48.4%

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

      3. associate-*r/ 48.4%

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

      4. metadata-eval 48.4%

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

      5. fma-neg 48.4%

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

      6. metadata-eval 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-/l* 48.4%

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

      3. +-commutative 48.4%

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

   8. Simplified 48.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-234}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \left(1 - {\varepsilon}^{2}\right) \cdot \frac{-1 + \frac{1}{\varepsilon}}{1 - \varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-281}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\frac{1 + \varepsilon}{1 + \frac{1}{\varepsilon}}} + \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 9: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps_m}\\ t_1 := 1 - \frac{1}{eps_m}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\frac{2 + x \cdot \left(t_0 \cdot \left(eps_m - 1\right) + \frac{t_1 \cdot \left(-1 + {eps_m}^{2}\right)}{1 - eps_m}\right)}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-281}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(eps_m, eps_m, -1\right)}{\frac{1 + eps_m}{t_0}} + t_1 \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))) (t_1 (- 1.0 (/ 1.0 eps_m))))
   (if (<= x -7.5e+34)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x -2e-232)
       (/
        (+
         2.0
         (*
          x
          (+
           (* t_0 (- eps_m 1.0))
           (/ (* t_1 (+ -1.0 (pow eps_m 2.0))) (- 1.0 eps_m)))))
        2.0)
       (if (<= x 2.5e-281)
         1.0
         (/
          (+
           2.0
           (*
            x
            (+
             (/ (fma eps_m eps_m -1.0) (/ (+ 1.0 eps_m) t_0))
             (* t_1 (- -1.0 eps_m)))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = 1.0 - (1.0 / eps_m);
	double tmp;
	if (x <= -7.5e+34) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -2e-232) {
		tmp = (2.0 + (x * ((t_0 * (eps_m - 1.0)) + ((t_1 * (-1.0 + pow(eps_m, 2.0))) / (1.0 - eps_m))))) / 2.0;
	} else if (x <= 2.5e-281) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (x * ((fma(eps_m, eps_m, -1.0) / ((1.0 + eps_m) / t_0)) + (t_1 * (-1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(1.0 - Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -7.5e+34)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -2e-232)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(t_0 * Float64(eps_m - 1.0)) + Float64(Float64(t_1 * Float64(-1.0 + (eps_m ^ 2.0))) / Float64(1.0 - eps_m))))) / 2.0);
	elseif (x <= 2.5e-281)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(fma(eps_m, eps_m, -1.0) / Float64(Float64(1.0 + eps_m) / t_0)) + Float64(t_1 * Float64(-1.0 - eps_m))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+34], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2e-232], N[(N[(2.0 + N[(x * N[(N[(t$95$0 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(-1.0 + N[Power[eps$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.5e-281], 1.0, N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(N[(1.0 + eps$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.49999999999999976e34

   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

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   if -7.49999999999999976e34 < x < -2.00000000000000005e-232

   1. Initial program 59.0%

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

   3. Simplified 50.7%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. flip-+ 82.2%

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

      2. associate-*l/ 82.2%

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

      3. metadata-eval 82.2%

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

      4. pow2 82.2%

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

   6. Applied egg-rr 82.2%

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

   if -2.00000000000000005e-232 < x < 2.4999999999999999e-281

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 2.4999999999999999e-281 < x

   1. Initial program 75.8%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in x around 0 38.3%

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

      1. flip-- 48.4%

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

      2. +-commutative 48.4%

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

      3. associate-*r/ 48.4%

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

      4. metadata-eval 48.4%

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

      5. fma-neg 48.4%

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

      6. metadata-eval 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-/l* 48.4%

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

      3. +-commutative 48.4%

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

   8. Simplified 48.4%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-232}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \frac{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(-1 + {\varepsilon}^{2}\right)}{1 - \varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-281}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\frac{1 + \varepsilon}{1 + \frac{1}{\varepsilon}}} + \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 10: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps_m}\\ t_1 := 1 - \frac{1}{eps_m}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2 + x \cdot \left(eps_m + \frac{-1 + {eps_m}^{2}}{\frac{1 - eps_m}{t_1}}\right)}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{2 + x \cdot \left(t_0 \cdot \left(eps_m - 1\right) + t_1 \cdot \left(-1 - eps_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(eps_m, eps_m, -1\right)}{\frac{1 + eps_m}{t_0}} + \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))) (t_1 (- 1.0 (/ 1.0 eps_m))))
   (if (<= x -7.5e+34)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x -3.5e-43)
       (/
        (+
         2.0
         (* x (+ eps_m (/ (+ -1.0 (pow eps_m 2.0)) (/ (- 1.0 eps_m) t_1)))))
        2.0)
       (if (<= x 9.8e-32)
         (/ (+ 2.0 (* x (+ (* t_0 (- eps_m 1.0)) (* t_1 (- -1.0 eps_m))))) 2.0)
         (/
          (+
           2.0
           (*
            x
            (+
             (/ (fma eps_m eps_m -1.0) (/ (+ 1.0 eps_m) t_0))
             (- -1.0 eps_m))))
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = 1.0 - (1.0 / eps_m);
	double tmp;
	if (x <= -7.5e+34) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -3.5e-43) {
		tmp = (2.0 + (x * (eps_m + ((-1.0 + pow(eps_m, 2.0)) / ((1.0 - eps_m) / t_1))))) / 2.0;
	} else if (x <= 9.8e-32) {
		tmp = (2.0 + (x * ((t_0 * (eps_m - 1.0)) + (t_1 * (-1.0 - eps_m))))) / 2.0;
	} else {
		tmp = (2.0 + (x * ((fma(eps_m, eps_m, -1.0) / ((1.0 + eps_m) / t_0)) + (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(1.0 - Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -7.5e+34)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -3.5e-43)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps_m + Float64(Float64(-1.0 + (eps_m ^ 2.0)) / Float64(Float64(1.0 - eps_m) / t_1))))) / 2.0);
	elseif (x <= 9.8e-32)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(t_0 * Float64(eps_m - 1.0)) + Float64(t_1 * Float64(-1.0 - eps_m))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(fma(eps_m, eps_m, -1.0) / Float64(Float64(1.0 + eps_m) / t_0)) + Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e+34], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -3.5e-43], N[(N[(2.0 + N[(x * N[(eps$95$m + N[(N[(-1.0 + N[Power[eps$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps$95$m), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.8e-32], N[(N[(2.0 + N[(x * N[(N[(t$95$0 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m * eps$95$m + -1.0), $MachinePrecision] / N[(N[(1.0 + eps$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.49999999999999976e34

   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

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   if -7.49999999999999976e34 < x < -3.49999999999999997e-43

   1. Initial program 70.9%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in x around 0 24.1%

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

   5. Taylor expanded in eps around inf 4.3%

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

      1. *-commutative 4.3%

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

      2. flip-+ 47.3%

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

      3. associate-*r/ 47.3%

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

      4. metadata-eval 47.3%

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

      5. pow2 47.3%

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

   7. Applied egg-rr 47.3%

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

      1. *-commutative 47.3%

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

      2. associate-/l* 47.3%

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

   9. Simplified 47.3%

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

   if -3.49999999999999997e-43 < x < 9.7999999999999996e-32

   1. Initial program 48.4%

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

   3. Simplified 41.0%

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

   4. Taylor expanded in x around 0 85.2%

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

   if 9.7999999999999996e-32 < x

   1. Initial program 100.0%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. flip-- 18.7%

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

      2. +-commutative 18.7%

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

      3. associate-*r/ 18.7%

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

      4. metadata-eval 18.7%

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

      5. fma-neg 18.7%

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

      6. metadata-eval 18.7%

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

   6. Applied egg-rr 18.7%

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

      1. *-commutative 18.7%

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

      2. associate-/l* 18.7%

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

      3. +-commutative 18.7%

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

   8. Simplified 18.7%

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

   9. Taylor expanded in eps around inf 18.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1 + {\varepsilon}^{2}}{\frac{1 - \varepsilon}{1 - \frac{1}{\varepsilon}}}\right)}{2}\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\frac{1 + \varepsilon}{1 + \frac{1}{\varepsilon}}} + \left(-1 - \varepsilon\right)\right)}{2}\\ \end{array} \]

Alternative 11: 62.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-58}:\\ \;\;\;\;\frac{2 + x \cdot \left(eps_m + \frac{-1 + {eps_m}^{2}}{\frac{1 - eps_m}{1 - \frac{1}{eps_m}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{eps_m}\right) \cdot \left(eps_m - 1\right) + \frac{\frac{1}{eps_m} + \left(-1 - eps_m\right)}{1 - eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7.5e+34)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x -9e-58)
     (/
      (+
       2.0
       (*
        x
        (+
         eps_m
         (/
          (+ -1.0 (pow eps_m 2.0))
          (/ (- 1.0 eps_m) (- 1.0 (/ 1.0 eps_m)))))))
      2.0)
     (/
      (+
       2.0
       (*
        x
        (+
         (* (+ 1.0 (/ 1.0 eps_m)) (- eps_m 1.0))
         (/ (+ (/ 1.0 eps_m) (- -1.0 eps_m)) (- 1.0 eps_m)))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7.5e+34) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= -9e-58) {
		tmp = (2.0 + (x * (eps_m + ((-1.0 + pow(eps_m, 2.0)) / ((1.0 - eps_m) / (1.0 - (1.0 / eps_m))))))) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (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 <= (-7.5d+34)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= (-9d-58)) then
        tmp = (2.0d0 + (x * (eps_m + (((-1.0d0) + (eps_m ** 2.0d0)) / ((1.0d0 - eps_m) / (1.0d0 - (1.0d0 / eps_m))))))) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((1.0d0 + (1.0d0 / eps_m)) * (eps_m - 1.0d0)) + (((1.0d0 / eps_m) + ((-1.0d0) - eps_m)) / (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 <= -7.5e+34) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= -9e-58) {
		tmp = (2.0 + (x * (eps_m + ((-1.0 + Math.pow(eps_m, 2.0)) / ((1.0 - eps_m) / (1.0 - (1.0 / eps_m))))))) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -7.5e+34:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= -9e-58:
		tmp = (2.0 + (x * (eps_m + ((-1.0 + math.pow(eps_m, 2.0)) / ((1.0 - eps_m) / (1.0 - (1.0 / eps_m))))))) / 2.0
	else:
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (1.0 - eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7.5e+34)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= -9e-58)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(eps_m + Float64(Float64(-1.0 + (eps_m ^ 2.0)) / Float64(Float64(1.0 - eps_m) / Float64(1.0 - Float64(1.0 / eps_m))))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(eps_m - 1.0)) + Float64(Float64(Float64(1.0 / eps_m) + Float64(-1.0 - eps_m)) / 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 <= -7.5e+34)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= -9e-58)
		tmp = (2.0 + (x * (eps_m + ((-1.0 + (eps_m ^ 2.0)) / ((1.0 - eps_m) / (1.0 - (1.0 / eps_m))))))) / 2.0;
	else
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (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, -7.5e+34], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -9e-58], N[(N[(2.0 + N[(x * N[(eps$95$m + N[(N[(-1.0 + N[Power[eps$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps$95$m), $MachinePrecision] / N[(1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.49999999999999976e34

   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

   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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   10. Simplified 100.0%

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

   if -7.49999999999999976e34 < x < -9.0000000000000006e-58

   1. Initial program 69.2%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in x around 0 26.6%

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

   5. Taylor expanded in eps around inf 4.1%

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

      1. *-commutative 4.1%

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

      2. flip-+ 47.6%

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

      3. associate-*r/ 47.6%

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

      4. metadata-eval 47.6%

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

      5. pow2 47.6%

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

   7. Applied egg-rr 47.6%

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

      1. *-commutative 47.6%

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

      2. associate-/l* 47.6%

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

   9. Simplified 47.6%

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

   if -9.0000000000000006e-58 < x

   1. Initial program 67.9%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in x around 0 54.5%

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

      1. flip-+ 54.7%

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

      2. associate-*l/ 54.7%

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

      3. metadata-eval 54.7%

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

      4. pow2 54.7%

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

   6. Applied egg-rr 54.7%

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

   7. Taylor expanded in eps around 0 58.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-58}:\\ \;\;\;\;\frac{2 + x \cdot \left(\varepsilon + \frac{-1 + {\varepsilon}^{2}}{\frac{1 - \varepsilon}{1 - \frac{1}{\varepsilon}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \frac{\frac{1}{\varepsilon} + \left(-1 - \varepsilon\right)}{1 - \varepsilon}\right)}{2}\\ \end{array} \]

Alternative 12: 64.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{eps_m}\right) \cdot \left(eps_m - 1\right) + \frac{\frac{1}{eps_m} + \left(-1 - eps_m\right)}{1 - eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.2e-180)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (- eps_m 1.0))
       (/ (+ (/ 1.0 eps_m) (- -1.0 eps_m)) (- 1.0 eps_m)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.2e-180) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (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 <= 1.2d-180) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((1.0d0 + (1.0d0 / eps_m)) * (eps_m - 1.0d0)) + (((1.0d0 / eps_m) + ((-1.0d0) - eps_m)) / (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 <= 1.2e-180) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.2e-180:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (1.0 - eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.2e-180)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(eps_m - 1.0)) + Float64(Float64(Float64(1.0 / eps_m) + Float64(-1.0 - eps_m)) / 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 <= 1.2e-180)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (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, 1.2e-180], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999999e-180

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   4. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in eps around inf 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in eps around 0 84.1%

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

      1. mul-1-neg 84.1%

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

   10. Simplified 84.1%

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

   if 1.1999999999999999e-180 < x

   1. Initial program 81.5%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in x around 0 25.3%

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

      1. flip-+ 31.3%

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

      2. associate-*l/ 31.3%

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

      3. metadata-eval 31.3%

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

      4. pow2 31.3%

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

   6. Applied egg-rr 31.3%

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

   7. Taylor expanded in eps around 0 32.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \frac{\frac{1}{\varepsilon} + \left(-1 - \varepsilon\right)}{1 - \varepsilon}\right)}{2}\\ \end{array} \]

Alternative 13: 58.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{2 - x \cdot eps_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{eps_m}\right) \cdot \left(eps_m - 1\right) + \frac{\frac{1}{eps_m} + \left(-1 - eps_m\right)}{1 - eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-294)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (- eps_m 1.0))
       (/ (+ (/ 1.0 eps_m) (- -1.0 eps_m)) (- 1.0 eps_m)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-294) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (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-294)) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((1.0d0 + (1.0d0 / eps_m)) * (eps_m - 1.0d0)) + (((1.0d0 / eps_m) + ((-1.0d0) - eps_m)) / (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-294) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (1.0 - eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-294:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (1.0 - eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-294)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(eps_m - 1.0)) + Float64(Float64(Float64(1.0 / eps_m) + Float64(-1.0 - eps_m)) / 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-294)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + (((1.0 / eps_m) + (-1.0 - eps_m)) / (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-294], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000002e-294

   1. Initial program 71.6%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in x around 0 44.3%

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

   5. Taylor expanded in eps around 0 52.6%

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

   6. Taylor expanded in eps around 0 52.6%

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

      1. mul-1-neg 52.6%

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

   8. Simplified 52.6%

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

   if -1.00000000000000002e-294 < x

   1. Initial program 73.6%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 44.6%

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

      1. flip-+ 44.2%

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

      2. associate-*l/ 44.2%

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

      3. metadata-eval 44.2%

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

      4. pow2 44.2%

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

   6. Applied egg-rr 44.2%

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

   7. Taylor expanded in eps around 0 49.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \frac{\frac{1}{\varepsilon} + \left(-1 - \varepsilon\right)}{1 - \varepsilon}\right)}{2}\\ \end{array} \]

Alternative 14: 58.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{2 - x \cdot eps_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{eps_m}\right) \cdot \left(eps_m - 1\right) + \left(1 + eps_m\right) \cdot \frac{1}{eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-294)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (- eps_m 1.0))
       (* (+ 1.0 eps_m) (/ 1.0 eps_m)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-294) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + ((1.0 + eps_m) * (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-294)) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((1.0d0 + (1.0d0 / eps_m)) * (eps_m - 1.0d0)) + ((1.0d0 + eps_m) * (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-294) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + ((1.0 + eps_m) * (1.0 / eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-294:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + ((1.0 + eps_m) * (1.0 / eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-294)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(eps_m - 1.0)) + Float64(Float64(1.0 + eps_m) * 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-294)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = (2.0 + (x * (((1.0 + (1.0 / eps_m)) * (eps_m - 1.0)) + ((1.0 + eps_m) * (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-294], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000002e-294

   1. Initial program 71.6%

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

   3. Simplified 67.0%

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

   4. Taylor expanded in x around 0 44.3%

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

   5. Taylor expanded in eps around 0 52.6%

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

   6. Taylor expanded in eps around 0 52.6%

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

      1. mul-1-neg 52.6%

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

   8. Simplified 52.6%

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

   if -1.00000000000000002e-294 < x

   1. Initial program 73.6%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 44.6%

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

      1. sub-neg 44.6%

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

      2. flip-+ 28.4%

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

      3. metadata-eval 28.4%

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

      4. distribute-neg-frac 28.4%

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

      5. metadata-eval 28.4%

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

      6. distribute-neg-frac 28.4%

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

      7. metadata-eval 28.4%

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

      8. distribute-neg-frac 28.4%

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

      9. metadata-eval 28.4%

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

   6. Applied egg-rr 28.4%

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

   7. Taylor expanded in eps around 0 49.6%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-294}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) + \left(1 + \varepsilon\right) \cdot \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 15: 50.7% accurate, 11.9× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 + (x * ((-1.0 / eps_m) + ((1.0 - (1.0 / eps_m)) * (-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 = (2.0d0 + (x * (((-1.0d0) / eps_m) + ((1.0d0 - (1.0d0 / eps_m)) * ((-1.0d0) - eps_m))))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Simplified 67.6%

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

4. Taylor expanded in x around 0 44.5%

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

5. Taylor expanded in eps around 0 49.1%

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

6. Final simplification 49.1%

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

Alternative 16: 50.7% accurate, 32.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 - x \cdot eps_m}{2} \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 - (x * 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 = (2.0d0 - (x * eps_m)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Simplified 67.6%

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

4. Taylor expanded in x around 0 44.5%

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

5. Taylor expanded in eps around 0 49.1%

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

6. Taylor expanded in eps around 0 49.1%

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

   1. mul-1-neg 49.1%

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

8. Simplified 49.1%

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

9. Final simplification 49.1%

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

Alternative 17: 44.1% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in x around 0 44.5%

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

5. Final simplification 44.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023332 
(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))