NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.9%

Time: 12.4s

Alternatives: 14

Speedup: 2.0×

• 37.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\left(1 - \varepsilon \cdot x\right) + -1}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps 2e-13)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (exp (+ (- 1.0 (* eps x)) -1.0))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps <= 2e-13) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp(((1.0 - (eps * x)) + -1.0))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps <= 2d-13) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp(((1.0d0 - (eps * x)) + (-1.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps <= 2e-13) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp(((1.0 - (eps * x)) + -1.0))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps <= 2e-13:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp(((1.0 - (eps * x)) + -1.0))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps <= 2e-13)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(Float64(1.0 - Float64(eps * x)) + -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps <= 2e-13)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp(((1.0 - (eps * x)) + -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 2e-13], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(1.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.0000000000000001e-13

   1. Initial program 64.5%

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

      1. div-sub 64.5%

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

      2. +-rgt-identity 64.5%

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

      3. div-sub 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in eps around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. distribute-lft1-in 66.0%

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

      3. mul-1-neg 66.0%

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

      4. distribute-lft-out 66.0%

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

      5. mul-1-neg 66.0%

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

      6. *-commutative 66.0%

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

      7. distribute-lft1-in 67.1%

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

      8. mul-1-neg 67.1%

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

   6. Simplified 67.1%

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

   if 2.0000000000000001e-13 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. distribute-rgt-neg-out 100.0%

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

      2. +-commutative 100.0%

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

      3. expm1-log1p-u 64.4%

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

      4. expm1-udef 64.4%

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

      5. log1p-udef 64.4%

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

      6. add-exp-log 100.0%

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

      7. unsub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\left(1 - \varepsilon \cdot x\right) + -1}}{2}\\ \end{array} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps 2e-13)
     (/ (+ t_0 t_0) 2.0)
     (/ (+ (exp (- (* eps x) x)) (exp (* x (- -1.0 eps)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps <= 2e-13) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp(((eps * x) - x)) + exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps <= 2d-13) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp(((eps * x) - x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps <= 2e-13) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp(((eps * x) - x)) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps <= 2e-13:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp(((eps * x) - x)) + math.exp((x * (-1.0 - eps)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps <= 2e-13)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps <= 2e-13)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp(((eps * x) - x)) + exp((x * (-1.0 - eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 2e-13], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.0000000000000001e-13

   1. Initial program 64.5%

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

      1. div-sub 64.5%

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

      2. +-rgt-identity 64.5%

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

      3. div-sub 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in eps around 0 66.0%

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

      1. *-commutative 66.0%

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

      2. distribute-lft1-in 66.0%

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

      3. mul-1-neg 66.0%

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

      4. distribute-lft-out 66.0%

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

      5. mul-1-neg 66.0%

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

      6. *-commutative 66.0%

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

      7. distribute-lft1-in 67.1%

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

      8. mul-1-neg 67.1%

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

   6. Simplified 67.1%

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

   if 2.0000000000000001e-13 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. distribute-rgt-neg-out 100.0%

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

      2. +-commutative 100.0%

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

      3. expm1-log1p-u 64.4%

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

      4. expm1-udef 64.4%

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

      5. log1p-udef 64.4%

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

      6. add-exp-log 100.0%

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

      7. unsub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 100.0%

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

       1. +-commutative 100.0%

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

       2. distribute-neg-in 100.0%

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

       3. neg-mul-1 100.0%

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

       4. +-commutative 100.0%

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

       5. sub-neg 100.0%

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

       6. distribute-rgt-out-- 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (- (* eps x) x)) (exp (* x (- -1.0 eps)))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp(((eps * x) - x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

   1. div-sub 74.6%

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

   2. +-rgt-identity 74.6%

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

   3. div-sub 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in eps around inf 98.1%

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

   1. mul-1-neg 98.1%

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

   2. *-commutative 98.1%

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

   3. mul-1-neg 98.1%

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

   4. associate-*r* 98.1%

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

   5. +-commutative 98.1%

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

   6. associate-*r* 98.1%

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

   7. mul-1-neg 98.1%

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

   8. *-commutative 98.1%

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

   9. distribute-rgt-neg-in 98.1%

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

   10. +-commutative 98.1%

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

6. Simplified 98.1%

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

   1. distribute-rgt-neg-out 98.1%

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

   2. +-commutative 98.1%

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

   3. expm1-log1p-u 60.2%

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

   4. expm1-udef 60.2%

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

   5. log1p-udef 60.2%

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

   6. add-exp-log 98.1%

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

   7. unsub-neg 98.1%

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

   8. distribute-rgt-in 98.1%

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

   9. *-un-lft-identity 98.1%

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

8. Applied egg-rr 98.1%

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

9. Taylor expanded in x around inf 98.1%

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

    1. +-commutative 98.1%

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

    2. distribute-neg-in 98.1%

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

    3. neg-mul-1 98.1%

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

    4. +-commutative 98.1%

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

    5. sub-neg 98.1%

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

    6. distribute-rgt-out-- 98.1%

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

11. Simplified 98.1%

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

12. Final simplification 98.1%

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

Alternative 4: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-250}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+100} \lor \neg \left(x \leq 5.8 \cdot 10^{+141}\right) \land \left(x \leq 10^{+253} \lor \neg \left(x \leq 3.8 \cdot 10^{+283}\right)\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-250)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (or (<= x 3.5e+100)
           (and (not (<= x 5.8e+141))
                (or (<= x 1e+253) (not (<= x 3.8e+283)))))
     (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)
     (/ (/ 2.0 (exp x)) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1e-250) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 3.5e+100) || (!(x <= 5.8e+141) && ((x <= 1e+253) || !(x <= 3.8e+283)))) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = (2.0 / exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1d-250)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 3.5d+100) .or. (.not. (x <= 5.8d+141)) .and. (x <= 1d+253) .or. (.not. (x <= 3.8d+283))) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = (2.0d0 / exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-250) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 3.5e+100) || (!(x <= 5.8e+141) && ((x <= 1e+253) || !(x <= 3.8e+283)))) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1e-250:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 3.5e+100) or (not (x <= 5.8e+141) and ((x <= 1e+253) or not (x <= 3.8e+283))):
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = (2.0 / math.exp(x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1e-250)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 3.5e+100) || (!(x <= 5.8e+141) && ((x <= 1e+253) || !(x <= 3.8e+283))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-250)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 3.5e+100) || (~((x <= 5.8e+141)) && ((x <= 1e+253) || ~((x <= 3.8e+283)))))
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = (2.0 / exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1e-250], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.5e+100], And[N[Not[LessEqual[x, 5.8e+141]], $MachinePrecision], Or[LessEqual[x, 1e+253], N[Not[LessEqual[x, 3.8e+283]], $MachinePrecision]]]], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0000000000000001e-250

   1. Initial program 76.0%

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

      1. div-sub 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. div-sub 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in x around 0 41.0%

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

   5. Taylor expanded in eps around inf 60.4%

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

      1. cancel-sign-sub-inv 60.4%

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

      2. metadata-eval 60.4%

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

      3. *-lft-identity 60.4%

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

      4. exp-prod 60.4%

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

      5. +-commutative 60.4%

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

      6. remove-double-neg 60.4%

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

      7. mul-1-neg 60.4%

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

      8. sub-neg 60.4%

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

      9. exp-prod 60.4%

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

      10. remove-double-neg 60.4%

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

      11. mul-1-neg 60.4%

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

      12. sub-neg 60.4%

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

      13. mul-1-neg 60.4%

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

      14. mul-1-neg 60.4%

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

      15. *-commutative 60.4%

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

      16. cancel-sign-sub-inv 60.4%

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

   7. Simplified 60.4%

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

   if -1.0000000000000001e-250 < x < 3.49999999999999976e100 or 5.80000000000000013e141 < x < 9.9999999999999994e252 or 3.8000000000000002e283 < x

   1. Initial program 70.0%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in eps around inf 99.0%

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

   5. Taylor expanded in x around 0 68.7%

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

   if 3.49999999999999976e100 < x < 5.80000000000000013e141 or 9.9999999999999994e252 < x < 3.8000000000000002e283

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. distribute-rgt-neg-out 100.0%

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

      2. +-commutative 100.0%

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

      3. expm1-log1p-u 5.0%

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

      4. expm1-udef 5.0%

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

      5. log1p-udef 5.0%

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

      6. add-exp-log 100.0%

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

      7. unsub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around 0 80.3%

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

       1. exp-neg 80.3%

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

       2. associate-*r/ 80.3%

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

       3. metadata-eval 80.3%

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

   11. Simplified 80.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-250}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+100} \lor \neg \left(x \leq 5.8 \cdot 10^{+141}\right) \land \left(x \leq 10^{+253} \lor \neg \left(x \leq 3.8 \cdot 10^{+283}\right)\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \end{array} \]

Alternative 5: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := x \cdot \left(-1 - \varepsilon\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{-255}:\\ \;\;\;\;\frac{1 + e^{t_0}}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(1 + t_0\right) + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+142} \lor \neg \left(x \leq 6.3 \cdot 10^{+252}\right) \land x \leq 3.8 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (- -1.0 eps))))
   (if (<= x -3e-255)
     (/ (+ 1.0 (exp t_0)) 2.0)
     (if (<= x 3.7e+72)
       (/ (+ (+ 1.0 t_0) (exp (* eps x))) 2.0)
       (if (or (<= x 2.3e+142) (and (not (<= x 6.3e+252)) (<= x 3.8e+283)))
         (/ (/ 2.0 (exp x)) 2.0)
         (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0))))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = x * (-1.0 - eps);
	double tmp;
	if (x <= -3e-255) {
		tmp = (1.0 + exp(t_0)) / 2.0;
	} else if (x <= 3.7e+72) {
		tmp = ((1.0 + t_0) + exp((eps * x))) / 2.0;
	} else if ((x <= 2.3e+142) || (!(x <= 6.3e+252) && (x <= 3.8e+283))) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) - eps)
    if (x <= (-3d-255)) then
        tmp = (1.0d0 + exp(t_0)) / 2.0d0
    else if (x <= 3.7d+72) then
        tmp = ((1.0d0 + t_0) + exp((eps * x))) / 2.0d0
    else if ((x <= 2.3d+142) .or. (.not. (x <= 6.3d+252)) .and. (x <= 3.8d+283)) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = x * (-1.0 - eps);
	double tmp;
	if (x <= -3e-255) {
		tmp = (1.0 + Math.exp(t_0)) / 2.0;
	} else if (x <= 3.7e+72) {
		tmp = ((1.0 + t_0) + Math.exp((eps * x))) / 2.0;
	} else if ((x <= 2.3e+142) || (!(x <= 6.3e+252) && (x <= 3.8e+283))) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = x * (-1.0 - eps)
	tmp = 0
	if x <= -3e-255:
		tmp = (1.0 + math.exp(t_0)) / 2.0
	elif x <= 3.7e+72:
		tmp = ((1.0 + t_0) + math.exp((eps * x))) / 2.0
	elif (x <= 2.3e+142) or (not (x <= 6.3e+252) and (x <= 3.8e+283)):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = Float64(x * Float64(-1.0 - eps))
	tmp = 0.0
	if (x <= -3e-255)
		tmp = Float64(Float64(1.0 + exp(t_0)) / 2.0);
	elseif (x <= 3.7e+72)
		tmp = Float64(Float64(Float64(1.0 + t_0) + exp(Float64(eps * x))) / 2.0);
	elseif ((x <= 2.3e+142) || (!(x <= 6.3e+252) && (x <= 3.8e+283)))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = x * (-1.0 - eps);
	tmp = 0.0;
	if (x <= -3e-255)
		tmp = (1.0 + exp(t_0)) / 2.0;
	elseif (x <= 3.7e+72)
		tmp = ((1.0 + t_0) + exp((eps * x))) / 2.0;
	elseif ((x <= 2.3e+142) || (~((x <= 6.3e+252)) && (x <= 3.8e+283)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e-255], N[(N[(1.0 + N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.7e+72], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.3e+142], And[N[Not[LessEqual[x, 6.3e+252]], $MachinePrecision], LessEqual[x, 3.8e+283]]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.00000000000000002e-255

   1. Initial program 76.0%

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

      1. div-sub 76.0%

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

      2. +-rgt-identity 76.0%

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

      3. div-sub 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in x around 0 41.0%

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

   5. Taylor expanded in eps around inf 60.4%

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

      1. cancel-sign-sub-inv 60.4%

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

      2. metadata-eval 60.4%

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

      3. *-lft-identity 60.4%

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

      4. exp-prod 60.4%

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

      5. +-commutative 60.4%

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

      6. remove-double-neg 60.4%

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

      7. mul-1-neg 60.4%

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

      8. sub-neg 60.4%

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

      9. exp-prod 60.4%

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

      10. remove-double-neg 60.4%

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

      11. mul-1-neg 60.4%

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

      12. sub-neg 60.4%

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

      13. mul-1-neg 60.4%

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

      14. mul-1-neg 60.4%

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

      15. *-commutative 60.4%

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

      16. cancel-sign-sub-inv 60.4%

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

   7. Simplified 60.4%

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

   if -3.00000000000000002e-255 < x < 3.7000000000000002e72

   1. Initial program 57.9%

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

   3. Simplified 51.5%

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

   4. Taylor expanded in eps around inf 98.5%

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

   5. Taylor expanded in x around 0 79.9%

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

      1. mul-1-neg 79.9%

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

      2. *-commutative 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in eps around inf 80.0%

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

      1. *-commutative 80.0%

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

   10. Simplified 80.0%

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

   if 3.7000000000000002e72 < x < 2.30000000000000002e142 or 6.3000000000000004e252 < x < 3.8000000000000002e283

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. +-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. distribute-rgt-neg-out 100.0%

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

      2. +-commutative 100.0%

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

      3. expm1-log1p-u 6.3%

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

      4. expm1-udef 6.3%

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

      5. log1p-udef 6.3%

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

      6. add-exp-log 100.0%

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

      7. unsub-neg 100.0%

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

      8. distribute-rgt-in 100.0%

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

      9. *-un-lft-identity 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in eps around 0 69.2%

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

       1. exp-neg 69.2%

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

       2. associate-*r/ 69.2%

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

       3. metadata-eval 69.2%

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

   11. Simplified 69.2%

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

   if 2.30000000000000002e142 < x < 6.3000000000000004e252 or 3.8000000000000002e283 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 41.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-255}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(-1 - \varepsilon\right)\right) + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+142} \lor \neg \left(x \leq 6.3 \cdot 10^{+252}\right) \land x \leq 3.8 \cdot 10^{+283}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 6: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 5.2e+127)
   (/ (/ 2.0 (exp x)) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 5.2e+127) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 5.2d+127) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 5.2e+127) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 5.2e+127:
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 5.2e+127)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 5.2e+127)
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 5.2e+127], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.2000000000000004e127

   1. Initial program 69.9%

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

      1. div-sub 69.9%

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

      2. +-rgt-identity 69.9%

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

      3. div-sub 69.9%

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

   3. Simplified 69.9%

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

   4. Taylor expanded in eps around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. *-commutative 97.7%

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

      3. mul-1-neg 97.7%

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

      4. associate-*r* 97.7%

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

      5. +-commutative 97.7%

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

      6. associate-*r* 97.7%

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

      7. mul-1-neg 97.7%

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

      8. *-commutative 97.7%

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

      9. distribute-rgt-neg-in 97.7%

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

      10. +-commutative 97.7%

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

   6. Simplified 97.7%

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

      1. distribute-rgt-neg-out 97.7%

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

      2. +-commutative 97.7%

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

      3. expm1-log1p-u 60.7%

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

      4. expm1-udef 60.7%

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

      5. log1p-udef 60.7%

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

      6. add-exp-log 97.7%

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

      7. unsub-neg 97.7%

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

      8. distribute-rgt-in 97.7%

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

      9. *-un-lft-identity 97.7%

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

   8. Applied egg-rr 97.7%

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

   9. Taylor expanded in eps around 0 77.2%

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

       1. exp-neg 77.2%

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

       2. associate-*r/ 77.2%

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

       3. metadata-eval 77.2%

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

   11. Simplified 77.2%

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

   if 5.2000000000000004e127 < 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 90.3%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 58.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.2 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 7: 70.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+165} \lor \neg \left(x \leq 1.6 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (or (<= x 1.2e+165) (not (<= x 1.6e+224)))
   (/ (/ 2.0 (exp x)) 2.0)
   (/
    (+ 2.0 (* x (+ (/ 1.0 eps) (* (- 1.0 eps) (+ -1.0 (/ -1.0 eps))))))
    2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if ((x <= 1.2e+165) || !(x <= 1.6e+224)) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= 1.2d+165) .or. (.not. (x <= 1.6d+224))) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps) + ((1.0d0 - eps) * ((-1.0d0) + ((-1.0d0) / eps)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if ((x <= 1.2e+165) || !(x <= 1.6e+224)) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if (x <= 1.2e+165) or not (x <= 1.6e+224):
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if ((x <= 1.2e+165) || !(x <= 1.6e+224))
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(1.0 - eps) * Float64(-1.0 + Float64(-1.0 / eps)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= 1.2e+165) || ~((x <= 1.6e+224)))
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[Or[LessEqual[x, 1.2e+165], N[Not[LessEqual[x, 1.6e+224]], $MachinePrecision]], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - eps), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e165 or 1.60000000000000007e224 < x

   1. Initial program 73.0%

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

      1. div-sub 73.0%

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

      2. +-rgt-identity 73.0%

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

      3. div-sub 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in eps around inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. *-commutative 97.9%

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

      3. mul-1-neg 97.9%

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

      4. associate-*r* 97.9%

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

      5. +-commutative 97.9%

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

      6. associate-*r* 97.9%

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

      7. mul-1-neg 97.9%

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

      8. *-commutative 97.9%

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

      9. distribute-rgt-neg-in 97.9%

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

      10. +-commutative 97.9%

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

   6. Simplified 97.9%

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

      1. distribute-rgt-neg-out 97.9%

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

      2. +-commutative 97.9%

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

      3. expm1-log1p-u 61.0%

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

      4. expm1-udef 61.0%

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

      5. log1p-udef 61.0%

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

      6. add-exp-log 97.9%

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

      7. unsub-neg 97.9%

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

      8. distribute-rgt-in 97.9%

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

      9. *-un-lft-identity 97.9%

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

   8. Applied egg-rr 97.9%

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

   9. Taylor expanded in eps around 0 74.0%

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

       1. exp-neg 74.0%

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

       2. associate-*r/ 74.0%

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

       3. metadata-eval 74.0%

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

   11. Simplified 74.0%

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

   if 1.2e165 < x < 1.60000000000000007e224

   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%

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

   4. Taylor expanded in x around 0 3.1%

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

   5. Taylor expanded in eps around 0 27.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{+165} \lor \neg \left(x \leq 1.6 \cdot 10^{+224}\right):\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]

Alternative 8: 64.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5:\\ \;\;\;\;\frac{2 + \left(\varepsilon \cdot x\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+164}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+225}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 8.5)
   (/ (+ 2.0 (* (* eps x) (+ -1.0 (/ 1.0 eps)))) 2.0)
   (if (<= x 3.8e+164)
     0.0
     (if (<= x 4e+225)
       (/
        (+ 2.0 (* x (+ (/ 1.0 eps) (* (- 1.0 eps) (+ -1.0 (/ -1.0 eps))))))
        2.0)
       0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 8.5) {
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0;
	} else if (x <= 3.8e+164) {
		tmp = 0.0;
	} else if (x <= 4e+225) {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 8.5d0) then
        tmp = (2.0d0 + ((eps * x) * ((-1.0d0) + (1.0d0 / eps)))) / 2.0d0
    else if (x <= 3.8d+164) then
        tmp = 0.0d0
    else if (x <= 4d+225) then
        tmp = (2.0d0 + (x * ((1.0d0 / eps) + ((1.0d0 - eps) * ((-1.0d0) + ((-1.0d0) / eps)))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 8.5) {
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0;
	} else if (x <= 3.8e+164) {
		tmp = 0.0;
	} else if (x <= 4e+225) {
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 8.5:
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0
	elif x <= 3.8e+164:
		tmp = 0.0
	elif x <= 4e+225:
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 8.5)
		tmp = Float64(Float64(2.0 + Float64(Float64(eps * x) * Float64(-1.0 + Float64(1.0 / eps)))) / 2.0);
	elseif (x <= 3.8e+164)
		tmp = 0.0;
	elseif (x <= 4e+225)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(1.0 - eps) * Float64(-1.0 + Float64(-1.0 / eps)))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.5)
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0;
	elseif (x <= 3.8e+164)
		tmp = 0.0;
	elseif (x <= 4e+225)
		tmp = (2.0 + (x * ((1.0 / eps) + ((1.0 - eps) * (-1.0 + (-1.0 / eps)))))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 8.5], N[(N[(2.0 + N[(N[(eps * x), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e+164], 0.0, If[LessEqual[x, 4e+225], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - eps), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.5

   1. Initial program 64.7%

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

      1. div-sub 64.7%

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

      2. +-rgt-identity 64.7%

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

      3. div-sub 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in x around 0 42.3%

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

   5. Taylor expanded in x around 0 47.7%

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

   6. Taylor expanded in eps around inf 61.9%

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

   if 8.5 < x < 3.80000000000000021e164 or 3.99999999999999971e225 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.1%

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

      1. div-sub 55.1%

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

      2. rec-exp 55.1%

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

      3. mul-1-neg 55.1%

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

      4. +-inverses 55.1%

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

   6. Simplified 55.1%

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

   if 3.80000000000000021e164 < x < 3.99999999999999971e225

   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%

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

   4. Taylor expanded in x around 0 3.1%

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

   5. Taylor expanded in eps around 0 27.5%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5:\\ \;\;\;\;\frac{2 + \left(\varepsilon \cdot x\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+164}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+225}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 64.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5:\\ \;\;\;\;\frac{2 + \left(\varepsilon \cdot x\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+162}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+225}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 8.5)
   (/ (+ 2.0 (* (* eps x) (+ -1.0 (/ 1.0 eps)))) 2.0)
   (if (<= x 2.8e+162) 0.0 (if (<= x 1.35e+225) (* (* eps x) 0.5) 0.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 8.5) {
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0;
	} else if (x <= 2.8e+162) {
		tmp = 0.0;
	} else if (x <= 1.35e+225) {
		tmp = (eps * x) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 8.5d0) then
        tmp = (2.0d0 + ((eps * x) * ((-1.0d0) + (1.0d0 / eps)))) / 2.0d0
    else if (x <= 2.8d+162) then
        tmp = 0.0d0
    else if (x <= 1.35d+225) then
        tmp = (eps * x) * 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 8.5) {
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0;
	} else if (x <= 2.8e+162) {
		tmp = 0.0;
	} else if (x <= 1.35e+225) {
		tmp = (eps * x) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 8.5:
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0
	elif x <= 2.8e+162:
		tmp = 0.0
	elif x <= 1.35e+225:
		tmp = (eps * x) * 0.5
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 8.5)
		tmp = Float64(Float64(2.0 + Float64(Float64(eps * x) * Float64(-1.0 + Float64(1.0 / eps)))) / 2.0);
	elseif (x <= 2.8e+162)
		tmp = 0.0;
	elseif (x <= 1.35e+225)
		tmp = Float64(Float64(eps * x) * 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.5)
		tmp = (2.0 + ((eps * x) * (-1.0 + (1.0 / eps)))) / 2.0;
	elseif (x <= 2.8e+162)
		tmp = 0.0;
	elseif (x <= 1.35e+225)
		tmp = (eps * x) * 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 8.5], N[(N[(2.0 + N[(N[(eps * x), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e+162], 0.0, If[LessEqual[x, 1.35e+225], N[(N[(eps * x), $MachinePrecision] * 0.5), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.5

   1. Initial program 64.7%

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

      1. div-sub 64.7%

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

      2. +-rgt-identity 64.7%

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

      3. div-sub 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in x around 0 42.3%

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

   5. Taylor expanded in x around 0 47.7%

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

   6. Taylor expanded in eps around inf 61.9%

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

   if 8.5 < x < 2.79999999999999991e162 or 1.3499999999999999e225 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.1%

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

      1. div-sub 55.1%

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

      2. rec-exp 55.1%

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

      3. mul-1-neg 55.1%

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

      4. +-inverses 55.1%

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

   6. Simplified 55.1%

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

   if 2.79999999999999991e162 < x < 1.3499999999999999e225

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 33.8%

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

   5. Taylor expanded in eps around inf 27.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 27.4%

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

      2. *-commutative 27.4%

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

   7. Simplified 27.4%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5:\\ \;\;\;\;\frac{2 + \left(\varepsilon \cdot x\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+162}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+225}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 63.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+163}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+221}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0)
   (* (* eps x) -0.5)
   (if (<= x 8.2e+29)
     1.0
     (if (<= x 7.5e+163) 0.0 (if (<= x 8e+221) (* (* eps x) 0.5) 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 8.2e+29) {
		tmp = 1.0;
	} else if (x <= 7.5e+163) {
		tmp = 0.0;
	} else if (x <= 8e+221) {
		tmp = (eps * x) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps * x) * (-0.5d0)
    else if (x <= 8.2d+29) then
        tmp = 1.0d0
    else if (x <= 7.5d+163) then
        tmp = 0.0d0
    else if (x <= 8d+221) then
        tmp = (eps * x) * 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 8.2e+29) {
		tmp = 1.0;
	} else if (x <= 7.5e+163) {
		tmp = 0.0;
	} else if (x <= 8e+221) {
		tmp = (eps * x) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (eps * x) * -0.5
	elif x <= 8.2e+29:
		tmp = 1.0
	elif x <= 7.5e+163:
		tmp = 0.0
	elif x <= 8e+221:
		tmp = (eps * x) * 0.5
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(eps * x) * -0.5);
	elseif (x <= 8.2e+29)
		tmp = 1.0;
	elseif (x <= 7.5e+163)
		tmp = 0.0;
	elseif (x <= 8e+221)
		tmp = Float64(Float64(eps * x) * 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (eps * x) * -0.5;
	elseif (x <= 8.2e+29)
		tmp = 1.0;
	elseif (x <= 7.5e+163)
		tmp = 0.0;
	elseif (x <= 8e+221)
		tmp = (eps * x) * 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(eps * x), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 8.2e+29], 1.0, If[LessEqual[x, 7.5e+163], 0.0, If[LessEqual[x, 8e+221], N[(N[(eps * x), $MachinePrecision] * 0.5), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 95.6%

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

      1. div-sub 95.6%

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

      2. +-rgt-identity 95.6%

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

      3. div-sub 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around 0 41.8%

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

   5. Taylor expanded in x around 0 23.4%

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

   6. Taylor expanded in eps around inf 23.3%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 23.3%

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

   8. Simplified 23.3%

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

   if -1 < x < 8.2000000000000007e29

   1. Initial program 55.6%

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

      1. div-sub 55.6%

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

      2. +-rgt-identity 55.6%

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

      3. div-sub 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in x around 0 43.7%

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

   5. Taylor expanded in x around 0 54.5%

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

   6. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{1} \]

   if 8.2000000000000007e29 < x < 7.50000000000000001e163 or 8.0000000000000004e221 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.1%

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

      1. div-sub 58.1%

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

      2. rec-exp 58.1%

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

      3. mul-1-neg 58.1%

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

      4. +-inverses 58.1%

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

   6. Simplified 58.1%

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

   if 7.50000000000000001e163 < x < 8.0000000000000004e221

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 33.8%

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

   5. Taylor expanded in eps around inf 27.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 27.4%

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

      2. *-commutative 27.4%

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

   7. Simplified 27.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+163}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+221}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 63.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2 + \left(\frac{x}{\varepsilon} - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+169}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+223}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0)
   (/ (+ 2.0 (- (/ x eps) (* eps x))) 2.0)
   (if (<= x 8.2e+29)
     1.0
     (if (<= x 3.6e+169) 0.0 (if (<= x 7.2e+223) (* (* eps x) 0.5) 0.0)))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 + ((x / eps) - (eps * x))) / 2.0;
	} else if (x <= 8.2e+29) {
		tmp = 1.0;
	} else if (x <= 3.6e+169) {
		tmp = 0.0;
	} else if (x <= 7.2e+223) {
		tmp = (eps * x) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (2.0d0 + ((x / eps) - (eps * x))) / 2.0d0
    else if (x <= 8.2d+29) then
        tmp = 1.0d0
    else if (x <= 3.6d+169) then
        tmp = 0.0d0
    else if (x <= 7.2d+223) then
        tmp = (eps * x) * 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 + ((x / eps) - (eps * x))) / 2.0;
	} else if (x <= 8.2e+29) {
		tmp = 1.0;
	} else if (x <= 3.6e+169) {
		tmp = 0.0;
	} else if (x <= 7.2e+223) {
		tmp = (eps * x) * 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (2.0 + ((x / eps) - (eps * x))) / 2.0
	elif x <= 8.2e+29:
		tmp = 1.0
	elif x <= 3.6e+169:
		tmp = 0.0
	elif x <= 7.2e+223:
		tmp = (eps * x) * 0.5
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 + Float64(Float64(x / eps) - Float64(eps * x))) / 2.0);
	elseif (x <= 8.2e+29)
		tmp = 1.0;
	elseif (x <= 3.6e+169)
		tmp = 0.0;
	elseif (x <= 7.2e+223)
		tmp = Float64(Float64(eps * x) * 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (2.0 + ((x / eps) - (eps * x))) / 2.0;
	elseif (x <= 8.2e+29)
		tmp = 1.0;
	elseif (x <= 3.6e+169)
		tmp = 0.0;
	elseif (x <= 7.2e+223)
		tmp = (eps * x) * 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(2.0 + N[(N[(x / eps), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.2e+29], 1.0, If[LessEqual[x, 3.6e+169], 0.0, If[LessEqual[x, 7.2e+223], N[(N[(eps * x), $MachinePrecision] * 0.5), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 95.6%

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

      1. div-sub 95.6%

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

      2. +-rgt-identity 95.6%

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

      3. div-sub 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around 0 41.8%

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

   5. Taylor expanded in x around 0 23.4%

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

   6. Taylor expanded in eps around 0 23.4%

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

      1. neg-mul-1 23.4%

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

      2. associate-+r+ 23.4%

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

      3. mul-1-neg 23.4%

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

      4. distribute-neg-in 23.4%

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

      5. distribute-lft1-in 23.4%

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

      6. *-commutative 23.4%

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

      7. distribute-rgt-neg-out 23.4%

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

      8. +-commutative 23.4%

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

      9. associate-+r+ 23.4%

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

   8. Simplified 23.4%

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

   if -1 < x < 8.2000000000000007e29

   1. Initial program 55.6%

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

      1. div-sub 55.6%

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

      2. +-rgt-identity 55.6%

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

      3. div-sub 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in x around 0 43.7%

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

   5. Taylor expanded in x around 0 54.5%

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

   6. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{1} \]

   if 8.2000000000000007e29 < x < 3.6000000000000001e169 or 7.19999999999999982e223 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.1%

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

      1. div-sub 58.1%

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

      2. rec-exp 58.1%

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

      3. mul-1-neg 58.1%

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

      4. +-inverses 58.1%

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

   6. Simplified 58.1%

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

   if 3.6000000000000001e169 < x < 7.19999999999999982e223

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 33.8%

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

   5. Taylor expanded in eps around inf 27.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 27.4%

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

      2. *-commutative 27.4%

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

   7. Simplified 27.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2 + \left(\frac{x}{\varepsilon} - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+169}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+223}:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 64.6% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0) (* (* eps x) -0.5) (if (<= x 8.2e+29) 1.0 0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 8.2e+29) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (eps * x) * (-0.5d0)
    else if (x <= 8.2d+29) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (eps * x) * -0.5;
	} else if (x <= 8.2e+29) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (eps * x) * -0.5
	elif x <= 8.2e+29:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(eps * x) * -0.5);
	elseif (x <= 8.2e+29)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (eps * x) * -0.5;
	elseif (x <= 8.2e+29)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(eps * x), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 8.2e+29], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 95.6%

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

      1. div-sub 95.6%

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

      2. +-rgt-identity 95.6%

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

      3. div-sub 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around 0 41.8%

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

   5. Taylor expanded in x around 0 23.4%

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

   6. Taylor expanded in eps around inf 23.3%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 23.3%

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

   8. Simplified 23.3%

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

   if -1 < x < 8.2000000000000007e29

   1. Initial program 55.6%

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

      1. div-sub 55.6%

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

      2. +-rgt-identity 55.6%

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

      3. div-sub 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in x around 0 43.7%

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

   5. Taylor expanded in x around 0 54.5%

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

   6. Taylor expanded in x around 0 73.9%

      \[\leadsto \color{blue}{1} \]

   if 8.2000000000000007e29 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.5%

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

      1. div-sub 51.5%

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

      2. rec-exp 51.5%

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

      3. mul-1-neg 51.5%

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

      4. +-inverses 51.5%

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

   6. Simplified 51.5%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(\varepsilon \cdot x\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 56.9% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 8.2e+29) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 8.2e+29) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 8.2d+29) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 8.2e+29) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 8.2e+29:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 8.2e+29)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.2e+29)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 8.2e+29], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.2000000000000007e29

   1. Initial program 65.2%

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

      1. div-sub 65.2%

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

      2. +-rgt-identity 65.2%

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

      3. div-sub 65.2%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in x around 0 43.2%

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

   5. Taylor expanded in x around 0 47.0%

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

   6. Taylor expanded in x around 0 56.9%

      \[\leadsto \color{blue}{1} \]

   if 8.2000000000000007e29 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.5%

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

      1. div-sub 51.5%

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

      2. rec-exp 51.5%

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

      3. mul-1-neg 51.5%

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

      4. +-inverses 51.5%

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

   6. Simplified 51.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+29}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 44.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 1 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

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

Python

eps = abs(eps)
def code(x, eps):
	return 1.0

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 1.0

Derivation

1. Initial program 74.6%

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

   1. div-sub 74.6%

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

   2. +-rgt-identity 74.6%

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

   3. div-sub 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in x around 0 38.0%

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

5. Taylor expanded in x around 0 37.5%

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

6. Taylor expanded in x around 0 42.4%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 42.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(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))