NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 100.0%

Time: 12.1s

Alternatives: 11

Speedup: 1.1×

• 37.1% 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: 73.1% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps_m \leq 0.0001:\\ \;\;\;\;\frac{t_0 \cdot \left(x + 2\right) + x \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) \cdot e^{x \cdot \left(eps_m + -1\right)} + e^{x \cdot \left(-1 - eps_m\right)} \cdot \left(\frac{-1}{eps_m} - -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 0.0001)
     (/ (+ (* t_0 (+ x 2.0)) (* x t_0)) 2.0)
     (/
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
       (* (exp (* x (- -1.0 eps_m))) (- (/ -1.0 eps_m) -1.0)))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 0.0001) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 0.0001d0) then
        tmp = ((t_0 * (x + 2.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (((-1.0d0) / eps_m) - (-1.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.0001) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) + (Math.exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.0001:
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) + (math.exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.0001)
		tmp = Float64(Float64(Float64(t_0 * Float64(x + 2.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(Float64(-1.0 / eps_m) - -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.0001)
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.0001], N[(N[(N[(t$95$0 * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.00000000000000005e-4

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in eps around 0 70.0%

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

      1. associate--r+ 70.0%

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

      2. associate-*r* 70.0%

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

      3. mul-1-neg 70.0%

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

      4. cancel-sign-sub 70.0%

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

      5. distribute-rgt1-in 70.0%

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

      6. distribute-rgt-out-- 71.6%

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

      7. mul-1-neg 71.6%

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

      8. mul-1-neg 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in x around inf 70.0%

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

      1. +-commutative 70.0%

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

      2. distribute-rgt-in 71.6%

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

   9. Simplified 71.6%

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

   if 1.00000000000000005e-4 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-out 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0001:\\ \;\;\;\;\frac{e^{-x} \cdot \left(x + 2\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \end{array} \]

Alternative 2: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps_m \leq 0.0002:\\ \;\;\;\;\frac{t_0 \cdot \left(x + 2\right) + x \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps_m + -1\right)} + e^{eps_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 0.0002)
     (/ (+ (* t_0 (+ x 2.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 0.0002) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 0.0002d0) then
        tmp = ((t_0 * (x + 2.0d0)) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + exp((eps_m * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.0002) {
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.0002:
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.0002)
		tmp = Float64(Float64(Float64(t_0 * Float64(x + 2.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.0002)
		tmp = ((t_0 * (x + 2.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.0002], N[(N[(N[(t$95$0 * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.0000000000000001e-4

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in eps around 0 70.0%

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

      1. associate--r+ 70.0%

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

      2. associate-*r* 70.0%

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

      3. mul-1-neg 70.0%

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

      4. cancel-sign-sub 70.0%

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

      5. distribute-rgt1-in 70.0%

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

      6. distribute-rgt-out-- 71.6%

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

      7. mul-1-neg 71.6%

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

      8. mul-1-neg 71.6%

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

   6. Simplified 71.6%

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

   7. Taylor expanded in x around inf 70.0%

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

      1. +-commutative 70.0%

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

      2. distribute-rgt-in 71.6%

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

   9. Simplified 71.6%

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

   if 2.0000000000000001e-4 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt-out 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0002:\\ \;\;\;\;\frac{e^{-x} \cdot \left(x + 2\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 3: 96.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{2 + x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-eps_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-180}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - {\left(1 - eps_m\right)}^{2}}{eps_m + -2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(eps_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4.4e-56)
   (/ (+ 2.0 (* x (log1p (expm1 (- eps_m))))) 2.0)
   (if (<= x -6e-180)
     (/ (+ 2.0 (* x (/ (- 1.0 (pow (- 1.0 eps_m) 2.0)) (+ eps_m -2.0)))) 2.0)
     (/ (+ (exp (- x)) (exp (* x (+ eps_m -1.0)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.4e-56) {
		tmp = (2.0 + (x * log1p(expm1(-eps_m)))) / 2.0;
	} else if (x <= -6e-180) {
		tmp = (2.0 + (x * ((1.0 - pow((1.0 - eps_m), 2.0)) / (eps_m + -2.0)))) / 2.0;
	} else {
		tmp = (exp(-x) + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.4e-56) {
		tmp = (2.0 + (x * Math.log1p(Math.expm1(-eps_m)))) / 2.0;
	} else if (x <= -6e-180) {
		tmp = (2.0 + (x * ((1.0 - Math.pow((1.0 - eps_m), 2.0)) / (eps_m + -2.0)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4.4e-56:
		tmp = (2.0 + (x * math.log1p(math.expm1(-eps_m)))) / 2.0
	elif x <= -6e-180:
		tmp = (2.0 + (x * ((1.0 - math.pow((1.0 - eps_m), 2.0)) / (eps_m + -2.0)))) / 2.0
	else:
		tmp = (math.exp(-x) + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4.4e-56)
		tmp = Float64(Float64(2.0 + Float64(x * log1p(expm1(Float64(-eps_m))))) / 2.0);
	elseif (x <= -6e-180)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - (Float64(1.0 - eps_m) ^ 2.0)) / Float64(eps_m + -2.0)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4.4e-56], N[(N[(2.0 + N[(x * N[Log[1 + N[(Exp[(-eps$95$m)] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -6e-180], N[(N[(2.0 + N[(x * N[(N[(1.0 - N[Power[N[(1.0 - eps$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(eps$95$m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.40000000000000008e-56

   1. Initial program 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in eps around inf 94.6%

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

   5. Taylor expanded in eps around 0 89.5%

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

   6. Taylor expanded in x around 0 35.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 16.2%

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

      3. mul-1-neg 16.2%

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

      4. mul-1-neg 16.2%

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

      5. sqr-neg 16.2%

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

      6. sqrt-unprod 16.2%

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

      7. add-sqr-sqrt 42.2%

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

      8. add-exp-log 16.2%

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

      9. expm1-udef 16.2%

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

      10. sub-neg 16.2%

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

      11. log1p-def 16.2%

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

      12. expm1-log1p-u 42.2%

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

      13. log1p-expm1-u 59.0%

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

   8. Applied egg-rr 59.0%

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

   if -4.40000000000000008e-56 < x < -6.0000000000000001e-180

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 81.4%

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

   6. Taylor expanded in x around 0 72.4%

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

      1. flip-- 39.8%

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

      2. associate-*r/ 39.8%

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

      3. metadata-eval 39.8%

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

      4. sub-neg 39.8%

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

      5. mul-1-neg 39.8%

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

      6. mul-1-neg 39.8%

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

      7. sqr-neg 39.8%

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

      8. pow2 39.8%

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

      9. metadata-eval 39.8%

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

      10. +-commutative 39.8%

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

      11. add-sqr-sqrt 6.6%

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

      12. sqrt-unprod 72.0%

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

      13. mul-1-neg 72.0%

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

      14. mul-1-neg 72.0%

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

      15. sqr-neg 72.0%

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

      16. sqrt-unprod 65.4%

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

      17. add-sqr-sqrt 86.8%

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

   8. Applied egg-rr 86.8%

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

      1. *-commutative 86.8%

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

      2. associate-/l* 72.1%

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

      3. associate-/r/ 86.8%

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

      4. *-lft-identity 86.8%

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

      5. metadata-eval 86.8%

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

      6. times-frac 86.8%

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

      7. neg-mul-1 86.8%

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

      8. neg-mul-1 86.8%

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

      9. neg-sub0 86.8%

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

      10. +-commutative 86.8%

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

      11. associate--r+ 86.8%

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

      12. metadata-eval 86.8%

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

      13. neg-sub0 86.8%

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

      14. associate-+r- 86.8%

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

      15. metadata-eval 86.8%

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

      16. associate--r- 86.8%

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

      17. metadata-eval 86.8%

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

      18. +-commutative 86.8%

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

   10. Simplified 86.8%

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

   if -6.0000000000000001e-180 < x

   1. Initial program 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in eps around inf 99.1%

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

   5. Taylor expanded in eps around 0 81.3%

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

   6. Taylor expanded in eps around -inf 81.3%

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

      1. cancel-sign-sub-inv 81.3%

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

      2. associate-*r* 81.3%

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

      3. neg-mul-1 81.3%

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

      4. mul-1-neg 81.3%

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

      5. sub-neg 81.3%

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

      6. *-commutative 81.3%

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

      7. metadata-eval 81.3%

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

      8. neg-mul-1 81.3%

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

      9. *-lft-identity 81.3%

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

   8. Simplified 81.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{2 + x \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(-\varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-180}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - {\left(1 - \varepsilon\right)}^{2}}{\varepsilon + -2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 4: 90.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps_m \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_0 \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + e^{eps_m \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 6.2e-14)
     (/ (* t_0 2.0) 2.0)
     (/ (+ t_0 (exp (* eps_m x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 6.2e-14) {
		tmp = (t_0 * 2.0) / 2.0;
	} else {
		tmp = (t_0 + exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 6.2d-14) then
        tmp = (t_0 * 2.0d0) / 2.0d0
    else
        tmp = (t_0 + exp((eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 6.2e-14) {
		tmp = (t_0 * 2.0) / 2.0;
	} else {
		tmp = (t_0 + Math.exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 6.2e-14:
		tmp = (t_0 * 2.0) / 2.0
	else:
		tmp = (t_0 + math.exp((eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 6.2e-14)
		tmp = Float64(Float64(t_0 * 2.0) / 2.0);
	else
		tmp = Float64(Float64(t_0 + exp(Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 6.2e-14)
		tmp = (t_0 * 2.0) / 2.0;
	else
		tmp = (t_0 + exp((eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 6.2e-14], N[(N[(t$95$0 * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.20000000000000009e-14

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in eps around inf 98.1%

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

   5. Taylor expanded in eps around 0 81.7%

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

   6. Taylor expanded in eps around 0 77.2%

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

      1. cancel-sign-sub-inv 77.2%

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

      2. neg-mul-1 77.2%

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

      3. metadata-eval 77.2%

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

      4. neg-mul-1 77.2%

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

      5. *-lft-identity 77.2%

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

      6. count-2 77.2%

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

   8. Simplified 77.2%

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

   if 6.20000000000000009e-14 < eps

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in eps around inf 98.6%

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

   5. Taylor expanded in eps around 0 86.6%

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

   6. Taylor expanded in eps around -inf 86.6%

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

      1. cancel-sign-sub-inv 86.6%

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

      2. associate-*r* 86.6%

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

      3. neg-mul-1 86.6%

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

      4. mul-1-neg 86.6%

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

      5. sub-neg 86.6%

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

      6. *-commutative 86.6%

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

      7. metadata-eval 86.6%

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

      8. neg-mul-1 86.6%

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

      9. *-lft-identity 86.6%

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

   8. Simplified 86.6%

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

   9. Taylor expanded in eps around inf 86.6%

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

       1. *-commutative 86.6%

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

   11. Simplified 86.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]

Alternative 5: 72.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 1.12 \cdot 10^{+210}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{elif}\;eps_m \leq 4.5 \cdot 10^{+263}:\\ \;\;\;\;\frac{2 + eps_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - {\left(1 - eps_m\right)}^{2}}{eps_m + -2}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.12e+210)
   (/ (* (exp (- x)) 2.0) 2.0)
   (if (<= eps_m 4.5e+263)
     (/ (+ 2.0 (* eps_m x)) 2.0)
     (/
      (+ 2.0 (* x (/ (- 1.0 (pow (- 1.0 eps_m) 2.0)) (+ eps_m -2.0))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.12e+210) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else if (eps_m <= 4.5e+263) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - pow((1.0 - eps_m), 2.0)) / (eps_m + -2.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.12d+210) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else if (eps_m <= 4.5d+263) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 - ((1.0d0 - eps_m) ** 2.0d0)) / (eps_m + (-2.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.12e+210) {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	} else if (eps_m <= 4.5e+263) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - Math.pow((1.0 - eps_m), 2.0)) / (eps_m + -2.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.12e+210:
		tmp = (math.exp(-x) * 2.0) / 2.0
	elif eps_m <= 4.5e+263:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 - math.pow((1.0 - eps_m), 2.0)) / (eps_m + -2.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.12e+210)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	elseif (eps_m <= 4.5e+263)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - (Float64(1.0 - eps_m) ^ 2.0)) / Float64(eps_m + -2.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.12e+210)
		tmp = (exp(-x) * 2.0) / 2.0;
	elseif (eps_m <= 4.5e+263)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 - ((1.0 - eps_m) ^ 2.0)) / (eps_m + -2.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.12e+210], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 4.5e+263], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 - N[Power[N[(1.0 - eps$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(eps$95$m + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.12000000000000005e210

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in eps around inf 98.1%

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

   5. Taylor expanded in eps around 0 84.9%

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

   6. Taylor expanded in eps around 0 73.4%

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

      1. cancel-sign-sub-inv 73.4%

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

      2. neg-mul-1 73.4%

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

      3. metadata-eval 73.4%

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

      4. neg-mul-1 73.4%

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

      5. *-lft-identity 73.4%

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

      6. count-2 73.4%

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

   8. Simplified 73.4%

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

   if 1.12000000000000005e210 < eps < 4.50000000000000014e263

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 76.4%

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

   6. Taylor expanded in x around 0 45.3%

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

   7. Taylor expanded in eps around inf 45.3%

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

   if 4.50000000000000014e263 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 46.2%

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

   6. Taylor expanded in x around 0 12.6%

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

      1. flip-- 33.3%

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

      2. associate-*r/ 33.3%

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

      3. metadata-eval 33.3%

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

      4. sub-neg 33.3%

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

      5. mul-1-neg 33.3%

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

      6. mul-1-neg 33.3%

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

      7. sqr-neg 33.3%

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

      8. pow2 33.3%

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

      9. metadata-eval 33.3%

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

      10. +-commutative 33.3%

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

      11. add-sqr-sqrt 33.3%

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

      12. sqrt-unprod 0.0%

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

      13. mul-1-neg 0.0%

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

      14. mul-1-neg 0.0%

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

      15. sqr-neg 0.0%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 66.7%

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

   8. Applied egg-rr 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-/l* 11.1%

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

      3. associate-/r/ 66.7%

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

      4. *-lft-identity 66.7%

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

      5. metadata-eval 66.7%

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

      6. times-frac 66.7%

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

      7. neg-mul-1 66.7%

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

      8. neg-mul-1 66.7%

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

      9. neg-sub0 66.7%

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

      10. +-commutative 66.7%

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

      11. associate--r+ 66.7%

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

      12. metadata-eval 66.7%

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

      13. neg-sub0 66.7%

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

      14. associate-+r- 66.7%

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

      15. metadata-eval 66.7%

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

      16. associate--r- 66.7%

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

      17. metadata-eval 66.7%

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

      18. +-commutative 66.7%

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

   10. Simplified 66.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.12 \cdot 10^{+210}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.5 \cdot 10^{+263}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - {\left(1 - \varepsilon\right)}^{2}}{\varepsilon + -2}}{2}\\ \end{array} \]

Alternative 6: 70.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 1.1 \cdot 10^{+210}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + eps_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.1e+210)
   (/ (* (exp (- x)) 2.0) 2.0)
   (/ (+ 2.0 (* eps_m x)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.1e+210) {
		tmp = (exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.1d+210) then
        tmp = (exp(-x) * 2.0d0) / 2.0d0
    else
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.1e+210) {
		tmp = (Math.exp(-x) * 2.0) / 2.0;
	} else {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.1e+210:
		tmp = (math.exp(-x) * 2.0) / 2.0
	else:
		tmp = (2.0 + (eps_m * x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.1e+210)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.1e+210)
		tmp = (exp(-x) * 2.0) / 2.0;
	else
		tmp = (2.0 + (eps_m * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.1e+210], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.09999999999999993e210

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in eps around inf 98.1%

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

   5. Taylor expanded in eps around 0 84.9%

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

   6. Taylor expanded in eps around 0 73.4%

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

      1. cancel-sign-sub-inv 73.4%

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

      2. neg-mul-1 73.4%

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

      3. metadata-eval 73.4%

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

      4. neg-mul-1 73.4%

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

      5. *-lft-identity 73.4%

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

      6. count-2 73.4%

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

   8. Simplified 73.4%

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

   if 1.09999999999999993e210 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 64.0%

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

   6. Taylor expanded in x around 0 31.9%

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

   7. Taylor expanded in eps around inf 31.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.1 \cdot 10^{+210}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 7: 57.6% accurate, 17.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+250}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+291}:\\ \;\;\;\;\frac{2 + eps_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.0)
   (/ (+ 2.0 (* x -2.0)) 2.0)
   (if (<= x 5.3e+250)
     0.0
     (if (<= x 2.5e+291) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.0) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else if (x <= 5.3e+250) {
		tmp = 0.0;
	} else if (x <= 2.5e+291) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (2.0d0 + (x * (-2.0d0))) / 2.0d0
    else if (x <= 5.3d+250) then
        tmp = 0.0d0
    else if (x <= 2.5d+291) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.0) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else if (x <= 5.3e+250) {
		tmp = 0.0;
	} else if (x <= 2.5e+291) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.0:
		tmp = (2.0 + (x * -2.0)) / 2.0
	elif x <= 5.3e+250:
		tmp = 0.0
	elif x <= 2.5e+291:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(2.0 + Float64(x * -2.0)) / 2.0);
	elseif (x <= 5.3e+250)
		tmp = 0.0;
	elseif (x <= 2.5e+291)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = (2.0 + (x * -2.0)) / 2.0;
	elseif (x <= 5.3e+250)
		tmp = 0.0;
	elseif (x <= 2.5e+291)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.0], N[(N[(2.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.3e+250], 0.0, If[LessEqual[x, 2.5e+291], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 97.9%

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

   5. Taylor expanded in eps around 0 87.7%

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

   6. Taylor expanded in x around 0 63.9%

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

   7. Taylor expanded in eps around 0 59.9%

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

   if 1 < x < 5.2999999999999999e250 or 2.5e291 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in eps around inf 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around 0 55.9%

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

      1. div-sub 55.9%

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

      2. neg-mul-1 55.9%

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

      3. +-inverses 55.9%

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

   9. Simplified 55.9%

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

   if 5.2999999999999999e250 < x < 2.5e291

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 73.2%

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

   6. Taylor expanded in x around 0 46.5%

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

   7. Taylor expanded in eps around inf 46.6%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+250}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+291}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 64.5% accurate, 17.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{2 - eps_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+249}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+291}:\\ \;\;\;\;\frac{2 + eps_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 230.0)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (if (<= x 1.9e+249)
     0.0
     (if (<= x 8.5e+291) (/ (+ 2.0 (* eps_m x)) 2.0) 0.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 230.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1.9e+249) {
		tmp = 0.0;
	} else if (x <= 8.5e+291) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 230.0d0) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 1.9d+249) then
        tmp = 0.0d0
    else if (x <= 8.5d+291) then
        tmp = (2.0d0 + (eps_m * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 230.0) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 1.9e+249) {
		tmp = 0.0;
	} else if (x <= 8.5e+291) {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 230.0:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 1.9e+249:
		tmp = 0.0
	elif x <= 8.5e+291:
		tmp = (2.0 + (eps_m * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 230.0)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 1.9e+249)
		tmp = 0.0;
	elseif (x <= 8.5e+291)
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 230.0)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 1.9e+249)
		tmp = 0.0;
	elseif (x <= 8.5e+291)
		tmp = (2.0 + (eps_m * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 230.0], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.9e+249], 0.0, If[LessEqual[x, 8.5e+291], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 230

   1. Initial program 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in eps around inf 97.5%

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

   5. Taylor expanded in eps around 0 87.3%

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

   6. Taylor expanded in x around 0 63.6%

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

      1. sub-neg 63.6%

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

      2. metadata-eval 63.6%

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

      3. +-commutative 63.6%

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

      4. add-sqr-sqrt 7.5%

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

      5. sqrt-unprod 60.6%

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

      6. mul-1-neg 60.6%

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

      7. mul-1-neg 60.6%

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

      8. sqr-neg 60.6%

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

      9. sqrt-unprod 50.5%

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

      10. add-sqr-sqrt 66.1%

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

   8. Applied egg-rr 66.1%

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

      1. associate-+r- 66.1%

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

      2. metadata-eval 66.1%

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

      3. neg-sub0 66.1%

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

   10. Simplified 66.1%

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

   if 230 < x < 1.8999999999999999e249 or 8.5000000000000003e291 < 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%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 56.7%

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

      1. div-sub 56.7%

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

      2. neg-mul-1 56.7%

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

      3. +-inverses 56.7%

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

   9. Simplified 56.7%

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

   if 1.8999999999999999e249 < x < 8.5000000000000003e291

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in eps around 0 73.2%

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

   6. Taylor expanded in x around 0 46.5%

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

   7. Taylor expanded in eps around inf 46.6%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+249}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+291}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 58.4% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.0) (/ (+ 2.0 (* x -2.0)) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.0) {
		tmp = (2.0 + (x * -2.0)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (2.0d0 + (x * (-2.0d0))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.0:
		tmp = (2.0 + (x * -2.0)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.0], N[(N[(2.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 97.9%

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

   5. Taylor expanded in eps around 0 87.7%

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

   6. Taylor expanded in x around 0 63.9%

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

   7. Taylor expanded in eps around 0 59.9%

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

   if 1 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in eps around inf 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around 0 50.8%

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

      1. div-sub 50.8%

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

      2. neg-mul-1 50.8%

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

      3. +-inverses 50.8%

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

   9. Simplified 50.8%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{2 + x \cdot -2}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 58.5% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 3350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 3350.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 3350.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 3350.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 3350.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 3350.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 3350.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 3350.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 3350.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3350

   1. Initial program 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in x around 0 58.8%

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

   if 3350 < 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%

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

   4. Taylor expanded in eps around inf 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 52.1%

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

      1. div-sub 52.1%

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

      2. neg-mul-1 52.1%

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

      3. +-inverses 52.1%

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

   9. Simplified 52.1%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 16.7% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 73.4%

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

3. Simplified 73.4%

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

4. Taylor expanded in eps around inf 53.1%

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

6. Simplified 66.7%

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

7. Taylor expanded in eps around 0 17.0%

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

   1. div-sub 17.0%

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

   2. neg-mul-1 17.0%

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

   3. +-inverses 17.2%

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

9. Simplified 17.2%

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

10. Final simplification 17.2%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023333 
(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))