NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 98.8%

Time: 20.0s

Alternatives: 16

Speedup: 2.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4e-277)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4e-277) {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp(x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 4d-277) then
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 4e-277) {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4e-277:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp(x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4e-277)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 4e-277)
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4e-277], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999988e-277

   1. Initial program 61.7%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 99.9%

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

       1. rec-exp 99.9%

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

       2. distribute-rgt-out-- 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   14. Simplified 100.0%

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

   if 3.99999999999999988e-277 < x

   1. Initial program 83.9%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around 0 77.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-277}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.102:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.102)
   (/ (+ (+ x 1.0) (- 1.0 x)) 2.0)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.102) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.102d0) then
        tmp = ((x + 1.0d0) + (1.0d0 - x)) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.102) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.102:
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.102)
		tmp = Float64(Float64(Float64(x + 1.0) + Float64(1.0 - x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.102)
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	else
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.102], N[(N[(N[(x + 1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.101999999999999993

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in x around 0 29.7%

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

   6. Taylor expanded in x around 0 20.4%

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

   7. Taylor expanded in eps around 0 66.9%

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

   if 0.101999999999999993 < eps

   1. Initial program 99.9%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 99.9%

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

       1. rec-exp 99.9%

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

       2. distribute-rgt-out-- 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in eps around inf 99.9%

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

       1. *-commutative 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.102:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 66.8%

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

5. Taylor expanded in eps around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 4: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-241}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+73} \lor \neg \left(x \leq 4 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \left(1 + x \cdot \left(eps\_m + -1\right)\right) + \frac{x + -1}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.75e-241)
   (/ (* 2.0 (exp (- x))) 2.0)
   (if (or (<= x 5e+73) (not (<= x 4e+158)))
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (/
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (* x (+ eps_m -1.0))))
       (/ (+ x -1.0) eps_m))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.75e-241) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if ((x <= 5e+73) || !(x <= 4e+158)) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.75d-241)) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if ((x <= 5d+73) .or. (.not. (x <= 4d+158))) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * (1.0d0 + (x * (eps_m + (-1.0d0))))) + ((x + (-1.0d0)) / eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.75e-241) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if ((x <= 5e+73) || !(x <= 4e+158)) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.75e-241:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif (x <= 5e+73) or not (x <= 4e+158):
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.75e-241)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif ((x <= 5e+73) || !(x <= 4e+158))
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(1.0 + Float64(x * Float64(eps_m + -1.0)))) + Float64(Float64(x + -1.0) / eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.75e-241)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif ((x <= 5e+73) || ~((x <= 4e+158)))
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.75e-241], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 5e+73], N[Not[LessEqual[x, 4e+158]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7499999999999999e-241

   1. Initial program 63.8%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around 0 92.0%

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

   7. Taylor expanded in eps around 0 83.6%

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

      1. neg-mul-1 83.6%

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

      2. rec-exp 83.6%

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

      3. count-2 83.6%

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

      4. rec-exp 83.6%

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

   9. Simplified 83.6%

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

   if -1.7499999999999999e-241 < x < 4.99999999999999976e73 or 3.99999999999999981e158 < x

   1. Initial program 76.8%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 88.5%

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

      1. *-commutative 88.5%

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

   8. Simplified 88.5%

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

   9. Taylor expanded in x around -inf 88.5%

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

       1. rec-exp 88.5%

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

       2. distribute-rgt-out-- 88.5%

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

   11. Simplified 88.5%

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

   12. Taylor expanded in x around 0 68.4%

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

   if 4.99999999999999976e73 < x < 3.99999999999999981e158

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.4%

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

   6. Taylor expanded in x around 0 33.9%

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

   7. Taylor expanded in eps around 0 46.8%

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

      1. mul-1-neg 46.8%

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

      2. unsub-neg 46.8%

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

   9. Simplified 46.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-241}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+73} \lor \neg \left(x \leq 4 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(\varepsilon + -1\right)\right) + \frac{x + -1}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+74} \lor \neg \left(x \leq 6.4 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \left(1 + x \cdot \left(eps\_m + -1\right)\right) + \frac{x + -1}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-240)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (or (<= x 3.7e+74) (not (<= x 6.4e+158)))
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (/
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (* x (+ eps_m -1.0))))
       (/ (+ x -1.0) eps_m))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-240) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 3.7e+74) || !(x <= 6.4e+158)) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2d-240)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if ((x <= 3.7d+74) .or. (.not. (x <= 6.4d+158))) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * (1.0d0 + (x * (eps_m + (-1.0d0))))) + ((x + (-1.0d0)) / eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-240) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 3.7e+74) || !(x <= 6.4e+158)) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-240:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif (x <= 3.7e+74) or not (x <= 6.4e+158):
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-240)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif ((x <= 3.7e+74) || !(x <= 6.4e+158))
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * Float64(1.0 + Float64(x * Float64(eps_m + -1.0)))) + Float64(Float64(x + -1.0) / eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-240)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif ((x <= 3.7e+74) || ~((x <= 6.4e+158)))
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps_m)) * (1.0 + (x * (eps_m + -1.0)))) + ((x + -1.0) / eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-240], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.7e+74], N[Not[LessEqual[x, 6.4e+158]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9999999999999999e-240

   1. Initial program 63.8%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 99.9%

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

       1. rec-exp 99.9%

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

       2. distribute-rgt-out-- 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around 0 70.3%

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

   if -1.9999999999999999e-240 < x < 3.7000000000000001e74 or 6.39999999999999989e158 < x

   1. Initial program 76.8%

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

   3. Simplified 73.3%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 88.5%

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

      1. *-commutative 88.5%

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

   8. Simplified 88.5%

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

   9. Taylor expanded in x around -inf 88.5%

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

       1. rec-exp 88.5%

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

       2. distribute-rgt-out-- 88.5%

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

   11. Simplified 88.5%

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

   12. Taylor expanded in x around 0 68.4%

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

   if 3.7000000000000001e74 < x < 6.39999999999999989e158

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.4%

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

   6. Taylor expanded in x around 0 33.9%

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

   7. Taylor expanded in eps around 0 46.8%

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

      1. mul-1-neg 46.8%

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

      2. unsub-neg 46.8%

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

   9. Simplified 46.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+74} \lor \neg \left(x \leq 6.4 \cdot 10^{+158}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(\varepsilon + -1\right)\right) + \frac{x + -1}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{1 + x \cdot \left(eps\_m - -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-241)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/
    (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (+ 1.0 (* x (- eps_m -1.0)))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-241) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (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) :: tmp
    if (x <= (-2d-241)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / (1.0d0 + (x * (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 tmp;
	if (x <= -2e-241) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m - -1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-241:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m - -1.0))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-241)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m - -1.0))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-241)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m - -1.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-241], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999999e-241

   1. Initial program 63.8%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 99.9%

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

       1. rec-exp 99.9%

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

       2. distribute-rgt-out-- 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around 0 70.3%

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

   if -1.9999999999999999e-241 < x

   1. Initial program 79.1%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 62.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-241}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon - -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-241}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.75e-241)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.75e-241) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (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) :: tmp
    if (x <= (-1.75d-241)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (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 tmp;
	if (x <= -1.75e-241) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + 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 <= -1.75e-241:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (1.0 + 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 <= -1.75e-241)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.75e-241)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.75e-241], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7499999999999999e-241

   1. Initial program 63.8%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 99.9%

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

       1. rec-exp 99.9%

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

       2. distribute-rgt-out-- 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around 0 70.3%

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

   if -1.7499999999999999e-241 < x

   1. Initial program 79.1%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 62.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-241}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 4.1 \cdot 10^{+259}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 4.1e+259)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 4.1e+259) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 4.1d+259) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 4.1e+259) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 4.1e+259:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 4.1e+259)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 4.1e+259)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 4.1e+259], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.1000000000000003e259

   1. Initial program 71.7%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in eps around 0 84.8%

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

   7. Taylor expanded in eps around 0 74.1%

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

      1. neg-mul-1 74.1%

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

      2. rec-exp 74.1%

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

      3. count-2 74.1%

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

      4. rec-exp 74.1%

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

   9. Simplified 74.1%

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

   if 4.1000000000000003e259 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 13.9%

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

   6. Taylor expanded in eps around 0 56.3%

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

   7. Taylor expanded in eps around 0 56.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.1 \cdot 10^{+259}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := x \cdot \left(eps\_m + -1\right)\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{t\_0 \cdot \left(1 - t\_1\right) - \left(-1 - \frac{-1}{eps\_m}\right) \cdot \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{t\_0 \cdot \left(1 + t\_1\right) + \frac{-1 - \left(eps\_m \cdot \left(\left(x \cdot eps\_m + \left(x + -1\right)\right) - x\right) - x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+268}:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{eps\_m} + t\_0 \cdot \left(eps\_m + -1\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps_m))) (t_1 (* x (+ eps_m -1.0))))
   (if (<= x -1.3e+107)
     (/
      (-
       (* t_0 (- 1.0 t_1))
       (* (- -1.0 (/ -1.0 eps_m)) (+ 1.0 (* x (- -1.0 eps_m)))))
      2.0)
     (if (<= x -9.2e-8)
       (/
        (+
         (* t_0 (+ 1.0 t_1))
         (/ (- -1.0 (- (* eps_m (- (+ (* x eps_m) (+ x -1.0)) x)) x)) eps_m))
        2.0)
       (if (<= x 5.7e+268)
         (/ (+ (+ x 1.0) (- 1.0 x)) 2.0)
         (/ (+ 2.0 (* x (+ (/ 1.0 eps_m) (* t_0 (+ eps_m -1.0))))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (1.0 / eps_m);
	double t_1 = x * (eps_m + -1.0);
	double tmp;
	if (x <= -1.3e+107) {
		tmp = ((t_0 * (1.0 - t_1)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0;
	} else if (x <= -9.2e-8) {
		tmp = ((t_0 * (1.0 + t_1)) + ((-1.0 - ((eps_m * (((x * eps_m) + (x + -1.0)) - x)) - x)) / eps_m)) / 2.0;
	} else if (x <= 5.7e+268) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (t_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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (1.0d0 / eps_m)
    t_1 = x * (eps_m + (-1.0d0))
    if (x <= (-1.3d+107)) then
        tmp = ((t_0 * (1.0d0 - t_1)) - (((-1.0d0) - ((-1.0d0) / eps_m)) * (1.0d0 + (x * ((-1.0d0) - eps_m))))) / 2.0d0
    else if (x <= (-9.2d-8)) then
        tmp = ((t_0 * (1.0d0 + t_1)) + (((-1.0d0) - ((eps_m * (((x * eps_m) + (x + (-1.0d0))) - x)) - x)) / eps_m)) / 2.0d0
    else if (x <= 5.7d+268) then
        tmp = ((x + 1.0d0) + (1.0d0 - x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 / eps_m) + (t_0 * (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 = 1.0 + (1.0 / eps_m);
	double t_1 = x * (eps_m + -1.0);
	double tmp;
	if (x <= -1.3e+107) {
		tmp = ((t_0 * (1.0 - t_1)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0;
	} else if (x <= -9.2e-8) {
		tmp = ((t_0 * (1.0 + t_1)) + ((-1.0 - ((eps_m * (((x * eps_m) + (x + -1.0)) - x)) - x)) / eps_m)) / 2.0;
	} else if (x <= 5.7e+268) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 / eps_m) + (t_0 * (eps_m + -1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (1.0 / eps_m)
	t_1 = x * (eps_m + -1.0)
	tmp = 0
	if x <= -1.3e+107:
		tmp = ((t_0 * (1.0 - t_1)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0
	elif x <= -9.2e-8:
		tmp = ((t_0 * (1.0 + t_1)) + ((-1.0 - ((eps_m * (((x * eps_m) + (x + -1.0)) - x)) - x)) / eps_m)) / 2.0
	elif x <= 5.7e+268:
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 / eps_m) + (t_0 * (eps_m + -1.0))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(1.0 / eps_m))
	t_1 = Float64(x * Float64(eps_m + -1.0))
	tmp = 0.0
	if (x <= -1.3e+107)
		tmp = Float64(Float64(Float64(t_0 * Float64(1.0 - t_1)) - Float64(Float64(-1.0 - Float64(-1.0 / eps_m)) * Float64(1.0 + Float64(x * Float64(-1.0 - eps_m))))) / 2.0);
	elseif (x <= -9.2e-8)
		tmp = Float64(Float64(Float64(t_0 * Float64(1.0 + t_1)) + Float64(Float64(-1.0 - Float64(Float64(eps_m * Float64(Float64(Float64(x * eps_m) + Float64(x + -1.0)) - x)) - x)) / eps_m)) / 2.0);
	elseif (x <= 5.7e+268)
		tmp = Float64(Float64(Float64(x + 1.0) + Float64(1.0 - x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 / eps_m) + Float64(t_0 * Float64(eps_m + -1.0))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (1.0 / eps_m);
	t_1 = x * (eps_m + -1.0);
	tmp = 0.0;
	if (x <= -1.3e+107)
		tmp = ((t_0 * (1.0 - t_1)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0;
	elseif (x <= -9.2e-8)
		tmp = ((t_0 * (1.0 + t_1)) + ((-1.0 - ((eps_m * (((x * eps_m) + (x + -1.0)) - x)) - x)) / eps_m)) / 2.0;
	elseif (x <= 5.7e+268)
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 / eps_m) + (t_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[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+107], N[(N[(N[(t$95$0 * N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -9.2e-8], N[(N[(N[(t$95$0 * N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 - N[(N[(eps$95$m * N[(N[(N[(x * eps$95$m), $MachinePrecision] + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.7e+268], N[(N[(N[(x + 1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(t$95$0 * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3000000000000001e107

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 35.3%

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

   6. Taylor expanded in x around 0 0.6%

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

      1. distribute-rgt-in 0.6%

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

      2. *-un-lft-identity 0.6%

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

      3. add-sqr-sqrt 0.3%

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

      4. sqrt-unprod 52.6%

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

      5. mul-1-neg 52.6%

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

      6. mul-1-neg 52.6%

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

      7. sqr-neg 52.6%

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

      8. sqrt-unprod 25.8%

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

      9. add-sqr-sqrt 25.8%

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

   8. Applied egg-rr 25.8%

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

      1. *-lft-identity 25.8%

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

      2. distribute-rgt-in 25.8%

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

   10. Simplified 25.8%

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

   if -1.3000000000000001e107 < x < -9.2000000000000003e-8

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 60.3%

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

   6. Taylor expanded in x around 0 1.4%

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

   7. Taylor expanded in eps around 0 32.0%

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

   if -9.2000000000000003e-8 < x < 5.7e268

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in x around 0 37.9%

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

   6. Taylor expanded in x around 0 29.9%

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

   7. Taylor expanded in eps around 0 69.9%

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

   if 5.7e268 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in eps around 0 43.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - x \cdot \left(\varepsilon + -1\right)\right) - \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(\varepsilon + -1\right)\right) + \frac{-1 - \left(\varepsilon \cdot \left(\left(x \cdot \varepsilon + \left(x + -1\right)\right) - x\right) - x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+268}:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1}{\varepsilon} + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.12 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 4.6 \cdot 10^{+198}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(-1 - \frac{-1}{eps\_m}\right) \cdot \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.12e+24)
   (/ (+ (+ x 1.0) (- 1.0 x)) 2.0)
   (if (<= eps_m 4.6e+198)
     (/
      (-
       (+ 1.0 (/ 1.0 eps_m))
       (* (- -1.0 (/ -1.0 eps_m)) (+ 1.0 (* x (- -1.0 eps_m)))))
      2.0)
     (/ (+ 2.0 (* x eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.12e+24) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else if (eps_m <= 4.6e+198) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.12d+24) then
        tmp = ((x + 1.0d0) + (1.0d0 - x)) / 2.0d0
    else if (eps_m <= 4.6d+198) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - (((-1.0d0) - ((-1.0d0) / eps_m)) * (1.0d0 + (x * ((-1.0d0) - eps_m))))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.12e+24) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else if (eps_m <= 4.6e+198) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.12e+24:
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0
	elif eps_m <= 4.6e+198:
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.12e+24)
		tmp = Float64(Float64(Float64(x + 1.0) + Float64(1.0 - x)) / 2.0);
	elseif (eps_m <= 4.6e+198)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(-1.0 - Float64(-1.0 / eps_m)) * Float64(1.0 + Float64(x * Float64(-1.0 - eps_m))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.12e+24)
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	elseif (eps_m <= 4.6e+198)
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 - (-1.0 / eps_m)) * (1.0 + (x * (-1.0 - eps_m))))) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.12e+24], N[(N[(N[(x + 1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 4.6e+198], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.12e24

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 32.6%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in eps around 0 67.2%

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

   if 1.12e24 < eps < 4.6000000000000001e198

   1. Initial program 99.8%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.6%

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

   6. Taylor expanded in x around 0 43.7%

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

   7. Taylor expanded in x around 0 49.5%

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

   if 4.6000000000000001e198 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.4%

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

   6. Taylor expanded in eps around 0 47.1%

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

   7. Taylor expanded in eps around 0 47.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.12 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 4.6 \cdot 10^{+198}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 - \frac{-1}{\varepsilon}\right) \cdot \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{elif}\;eps\_m \leq 1.95 \cdot 10^{+197}:\\ \;\;\;\;\frac{2 - x \cdot \left(eps\_m - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.5e+24)
   (/ (+ (+ x 1.0) (- 1.0 x)) 2.0)
   (if (<= eps_m 1.95e+197)
     (/ (- 2.0 (* x (- eps_m -1.0))) 2.0)
     (/ (+ 2.0 (* x eps_m)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.5e+24) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else if (eps_m <= 1.95e+197) {
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.5d+24) then
        tmp = ((x + 1.0d0) + (1.0d0 - x)) / 2.0d0
    else if (eps_m <= 1.95d+197) then
        tmp = (2.0d0 - (x * (eps_m - (-1.0d0)))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.5e+24) {
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	} else if (eps_m <= 1.95e+197) {
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.5e+24:
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0
	elif eps_m <= 1.95e+197:
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.5e+24)
		tmp = Float64(Float64(Float64(x + 1.0) + Float64(1.0 - x)) / 2.0);
	elseif (eps_m <= 1.95e+197)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps_m - -1.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.5e+24)
		tmp = ((x + 1.0) + (1.0 - x)) / 2.0;
	elseif (eps_m <= 1.95e+197)
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.5e+24], N[(N[(N[(x + 1.0), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.95e+197], N[(N[(2.0 - N[(x * N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.49999999999999997e24

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 32.6%

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

   6. Taylor expanded in x around 0 22.7%

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

   7. Taylor expanded in eps around 0 67.2%

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

   if 1.49999999999999997e24 < eps < 1.95e197

   1. Initial program 99.8%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in eps around inf 99.8%

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

   6. Taylor expanded in eps around inf 99.8%

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

      1. *-commutative 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 85.7%

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

   10. Taylor expanded in x around 0 49.5%

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

   if 1.95e197 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 14.4%

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

   6. Taylor expanded in eps around 0 47.1%

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

   7. Taylor expanded in eps around 0 47.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\left(x + 1\right) + \left(1 - x\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.95 \cdot 10^{+197}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\frac{2 - x \cdot \left(eps\_m - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-240)
   (/ (- 2.0 (* x (- eps_m -1.0))) 2.0)
   (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-240) {
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2d-240)) then
        tmp = (2.0d0 - (x * (eps_m - (-1.0d0)))) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-240) {
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-240:
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-240)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps_m - -1.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-240)
		tmp = (2.0 - (x * (eps_m - -1.0))) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-240], N[(N[(2.0 - N[(x * N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9999999999999999e-240

   1. Initial program 63.8%

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

   3. Simplified 54.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 70.3%

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

   10. Taylor expanded in x around 0 53.6%

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

   if -1.9999999999999999e-240 < x

   1. Initial program 79.1%

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

   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. Add Preprocessing

   5. Taylor expanded in x around 0 47.0%

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

   6. Taylor expanded in eps around 0 53.1%

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

   7. Taylor expanded in eps around 0 53.0%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.7% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -9.6e-11) (/ (* x (- -1.0 eps_m)) 2.0) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.6e-11) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-9.6d-11)) then
        tmp = (x * ((-1.0d0) - eps_m)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.6e-11) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -9.6e-11:
		tmp = (x * (-1.0 - eps_m)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -9.6e-11)
		tmp = Float64(Float64(x * Float64(-1.0 - eps_m)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -9.6e-11)
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -9.6e-11], N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000005e-11

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 46.1%

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

   7. Taylor expanded in x around inf 16.9%

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

      1. mul-1-neg 16.9%

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

      2. distribute-rgt-neg-in 16.9%

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

      3. mul-1-neg 16.9%

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

      4. distribute-lft-in 16.9%

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

      5. metadata-eval 16.9%

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

      6. mul-1-neg 16.9%

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

      7. unsub-neg 16.9%

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

   9. Simplified 16.9%

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

   if -9.6000000000000005e-11 < x

   1. Initial program 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in x around 0 56.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.9% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -9.6e-11) (/ (* x (- -1.0 eps_m)) 2.0) (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.6e-11) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-9.6d-11)) then
        tmp = (x * ((-1.0d0) - eps_m)) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.6e-11) {
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -9.6e-11:
		tmp = (x * (-1.0 - eps_m)) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -9.6e-11)
		tmp = Float64(Float64(x * Float64(-1.0 - eps_m)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -9.6e-11)
		tmp = (x * (-1.0 - eps_m)) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -9.6e-11], N[(N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000005e-11

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 46.1%

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

   7. Taylor expanded in x around inf 16.9%

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

      1. mul-1-neg 16.9%

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

      2. distribute-rgt-neg-in 16.9%

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

      3. mul-1-neg 16.9%

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

      4. distribute-lft-in 16.9%

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

      5. metadata-eval 16.9%

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

      6. mul-1-neg 16.9%

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

      7. unsub-neg 16.9%

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

   9. Simplified 16.9%

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

   if -9.6000000000000005e-11 < x

   1. Initial program 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in x around 0 56.5%

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

   6. Taylor expanded in eps around 0 60.4%

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

   7. Taylor expanded in eps around 0 60.3%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.6% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot eps\_m}{-2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -9.6e-11) (/ (* x eps_m) (- 2.0)) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.6e-11) {
		tmp = (x * eps_m) / -2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-9.6d-11)) then
        tmp = (x * eps_m) / -2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -9.6e-11) {
		tmp = (x * eps_m) / -2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -9.6e-11:
		tmp = (x * eps_m) / -2.0
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -9.6e-11)
		tmp = Float64(Float64(x * eps_m) / Float64(-2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -9.6e-11)
		tmp = (x * eps_m) / -2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -9.6e-11], N[(N[(x * eps$95$m), $MachinePrecision] / (-2.0)), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -9.6000000000000005e-11

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 46.1%

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

   6. Taylor expanded in eps around inf 16.9%

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

      1. mul-1-neg 16.9%

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

      2. distribute-lft-neg-out 16.9%

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

      3. *-commutative 16.9%

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

   8. Simplified 16.9%

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

   if -9.6000000000000005e-11 < x

   1. Initial program 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in x around 0 56.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{-2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.6% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around 0 48.0%

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

6. Final simplification 48.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(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))