NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.9%

Time: 14.3s

Alternatives: 15

Speedup: 2.0×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-39}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{{e}^{\left(\mathsf{fma}\left(x, eps\_m, x\right)\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-39)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (pow E (fma x eps_m x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-39) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / pow(((double) M_E), fma(x, eps_m, x)))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-39)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / (exp(1) ^ fma(x, eps_m, x)))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-39], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[E, N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.9999999999999998e-39

   1. Initial program 62.0%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in eps around 0 35.1%

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

      1. associate-+r+ 72.5%

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

      2. mul-1-neg 72.5%

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

      3. sub-neg 72.5%

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

      4. +-inverses 72.5%

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

      5. distribute-lft-out 72.5%

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

      6. distribute-rgt1-in 73.1%

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

      7. mul-1-neg 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in eps around 0 73.1%

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

      1. +-commutative 73.1%

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

   10. Simplified 73.1%

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

   if 4.9999999999999998e-39 < eps

   1. Initial program 95.0%

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

   3. Simplified 84.7%

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

      1. *-un-lft-identity 99.9%

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

      2. exp-prod 99.9%

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

      3. +-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-define 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. exp-1-e 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-39}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{e}^{\left(\mathsf{fma}\left(x, \varepsilon, x\right)\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-39}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{e^{x + eps\_m \cdot x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-39)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (+ x (* eps_m x))))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-39) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5d-39) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / exp((x + (eps_m * x))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-39) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / Math.exp((x + (eps_m * x))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5e-39:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp((x + (eps_m * x))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-39)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(eps_m * x))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5e-39)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / exp((x + (eps_m * x))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-39], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.9999999999999998e-39

   1. Initial program 62.0%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in eps around 0 35.1%

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

      1. associate-+r+ 72.5%

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

      2. mul-1-neg 72.5%

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

      3. sub-neg 72.5%

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

      4. +-inverses 72.5%

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

      5. distribute-lft-out 72.5%

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

      6. distribute-rgt1-in 73.1%

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

      7. mul-1-neg 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in eps around 0 73.1%

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

      1. +-commutative 73.1%

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

   10. Simplified 73.1%

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

   if 4.9999999999999998e-39 < eps

   1. Initial program 95.0%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in eps around inf 99.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-39}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{x + eps\_m \cdot x}} + e^{eps\_m \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.35e-12)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (/ 1.0 (exp (+ x (* eps_m x)))) (exp (* eps_m x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / exp((x + (eps_m * x)))) + exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 2.35d-12) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = ((1.0d0 / exp((x + (eps_m * x)))) + exp((eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / Math.exp((x + (eps_m * x)))) + Math.exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2.35e-12:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = ((1.0 / math.exp((x + (eps_m * x)))) + math.exp((eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.35e-12)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / exp(Float64(x + Float64(eps_m * x)))) + exp(Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2.35e-12)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = ((1.0 / exp((x + (eps_m * x)))) + exp((eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.35e-12], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.34999999999999988e-12

   1. Initial program 60.9%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around 0 34.7%

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

      1. associate-+r+ 73.3%

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

      2. mul-1-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-inverses 73.3%

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

      5. distribute-lft-out 73.3%

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

      6. distribute-rgt1-in 73.9%

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

      7. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in eps around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 2.34999999999999988e-12 < 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 88.8%

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

   5. Taylor expanded in eps around inf 99.9%

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

   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%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{e^{x + \varepsilon \cdot x}} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{eps\_m \cdot x} + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.35e-12)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (exp (* eps_m x)) (exp (* x (- -1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (exp((eps_m * x)) + exp((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 (eps_m <= 2.35d-12) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (exp((eps_m * x)) + exp((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 (eps_m <= 2.35e-12) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (Math.exp((eps_m * x)) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2.35e-12:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (math.exp((eps_m * x)) + math.exp((x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.35e-12)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(exp(Float64(eps_m * x)) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2.35e-12)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (exp((eps_m * x)) + exp((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[eps$95$m, 2.35e-12], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.34999999999999988e-12

   1. Initial program 60.9%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around 0 34.7%

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

      1. associate-+r+ 73.3%

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

      2. mul-1-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-inverses 73.3%

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

      5. distribute-lft-out 73.3%

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

      6. distribute-rgt1-in 73.9%

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

      7. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in eps around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 2.34999999999999988e-12 < 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 88.8%

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

   5. Taylor expanded in eps around inf 99.9%

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

   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. *-commutative 99.9%

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

       2. neg-mul-1 99.9%

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

       3. fma-neg 99.9%

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

       4. remove-double-neg 99.9%

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

       5. *-rgt-identity 99.9%

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

       6. exp-neg 99.9%

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

       7. *-rgt-identity 99.9%

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

       8. fma-undefine 99.9%

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

       9. *-commutative 99.9%

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

       10. +-commutative 99.9%

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

       11. *-lft-identity 99.9%

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

       12. distribute-rgt-in 99.9%

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

       13. distribute-rgt-neg-in 99.9%

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

       14. distribute-neg-in 99.9%

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

       15. metadata-eval 99.9%

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

       16. unsub-neg 99.9%

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

   11. Simplified 99.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-228)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (<= x 5.8e+119) (/ (+ 1.0 (exp (* eps_m x))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-228) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 5.8e+119) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2d-228)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 5.8d+119) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-228) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 5.8e+119) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-228:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 5.8e+119:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-228)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 5.8e+119)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-228)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 5.8e+119)
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-228], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.8e+119], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000007e-228

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in x around 0 46.2%

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

   6. Taylor expanded in eps around inf 67.0%

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

      1. cancel-sign-sub-inv 67.0%

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

      2. metadata-eval 67.0%

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

      3. mul-1-neg 67.0%

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

      4. distribute-rgt-in 67.0%

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

      5. *-lft-identity 67.0%

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

      6. +-commutative 67.0%

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

      7. *-commutative 67.0%

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

      8. fma-undefine 67.0%

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

      9. *-lft-identity 67.0%

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

      10. fma-undefine 67.0%

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

      11. *-commutative 67.0%

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

      12. +-commutative 67.0%

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

      13. *-lft-identity 67.0%

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

      14. distribute-rgt-in 67.0%

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

      15. distribute-rgt-neg-in 67.0%

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

      16. distribute-neg-in 67.0%

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

      17. metadata-eval 67.0%

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

      18. unsub-neg 67.0%

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

   8. Simplified 67.0%

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

   9. Taylor expanded in eps around inf 67.9%

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

       1. neg-mul-1 67.9%

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

   11. Simplified 67.9%

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

   if -2.00000000000000007e-228 < x < 5.80000000000000014e119

   1. Initial program 60.6%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in eps around inf 90.0%

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

      1. *-commutative 90.0%

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

   8. Simplified 90.0%

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

   9. Taylor expanded in x around 0 76.6%

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

   if 5.80000000000000014e119 < 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%

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

   5. Taylor expanded in eps around 0 70.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 70.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 70.0%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 70.0%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 70.0%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 70.0%

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

   7. Simplified 70.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-228}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;t\_0 \cdot \left(x + 1\right)\\ \mathbf{elif}\;eps\_m \leq 5 \cdot 10^{+196}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 2.35e-12)
     (* t_0 (+ x 1.0))
     (if (<= eps_m 5e+196)
       (+ 1.0 (* x (- (* x 0.25) 0.5)))
       (/ (+ t_0 1.0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = t_0 * (x + 1.0);
	} else if (eps_m <= 5e+196) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 2.35d-12) then
        tmp = t_0 * (x + 1.0d0)
    else if (eps_m <= 5d+196) then
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    else
        tmp = (t_0 + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = t_0 * (x + 1.0);
	} else if (eps_m <= 5e+196) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = (t_0 + 1.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 2.35e-12:
		tmp = t_0 * (x + 1.0)
	elif eps_m <= 5e+196:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	else:
		tmp = (t_0 + 1.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 2.35e-12)
		tmp = Float64(t_0 * Float64(x + 1.0));
	elseif (eps_m <= 5e+196)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	else
		tmp = Float64(Float64(t_0 + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 2.35e-12)
		tmp = t_0 * (x + 1.0);
	elseif (eps_m <= 5e+196)
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	else
		tmp = (t_0 + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 2.35e-12], N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 5e+196], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 2.34999999999999988e-12

   1. Initial program 60.9%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around 0 34.7%

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

      1. associate-+r+ 73.3%

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

      2. mul-1-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-inverses 73.3%

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

      5. distribute-lft-out 73.3%

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

      6. distribute-rgt1-in 73.9%

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

      7. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in eps around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 2.34999999999999988e-12 < eps < 4.9999999999999998e196

   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 70.5%

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

   6. Taylor expanded in eps around inf 70.5%

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

      1. cancel-sign-sub-inv 70.5%

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

      2. metadata-eval 70.5%

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

      3. mul-1-neg 70.5%

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

      4. distribute-rgt-in 70.5%

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

      5. *-lft-identity 70.5%

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

      6. +-commutative 70.5%

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

      7. *-commutative 70.5%

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

      8. fma-undefine 70.5%

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

      9. *-lft-identity 70.5%

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

      10. fma-undefine 70.5%

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

      11. *-commutative 70.5%

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

      12. +-commutative 70.5%

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

      13. *-lft-identity 70.5%

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

      14. distribute-rgt-in 70.5%

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

      15. distribute-rgt-neg-in 70.5%

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

      16. distribute-neg-in 70.5%

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

      17. metadata-eval 70.5%

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

      18. unsub-neg 70.5%

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

   8. Simplified 70.5%

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

   9. Taylor expanded in eps around 0 57.8%

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

       1. mul-1-neg 57.8%

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

   11. Simplified 57.8%

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

   12. Taylor expanded in x around 0 64.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]

   if 4.9999999999999998e196 < 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 65.8%

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

   6. Taylor expanded in eps around inf 65.8%

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

      1. cancel-sign-sub-inv 65.8%

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

      2. metadata-eval 65.8%

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

      3. mul-1-neg 65.8%

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

      4. distribute-rgt-in 65.8%

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

      5. *-lft-identity 65.8%

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

      6. +-commutative 65.8%

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

      7. *-commutative 65.8%

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

      8. fma-undefine 65.8%

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

      9. *-lft-identity 65.8%

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

      10. fma-undefine 65.8%

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

      11. *-commutative 65.8%

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

      12. +-commutative 65.8%

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

      13. *-lft-identity 65.8%

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

      14. distribute-rgt-in 65.8%

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

      15. distribute-rgt-neg-in 65.8%

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

      16. distribute-neg-in 65.8%

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

      17. metadata-eval 65.8%

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

      18. unsub-neg 65.8%

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

   8. Simplified 65.8%

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

   9. Taylor expanded in eps around 0 60.1%

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

       1. mul-1-neg 60.1%

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

   11. Simplified 60.1%

       \[\leadsto \frac{\color{blue}{1 + e^{-x}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+196}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;t\_0 \cdot \left(x + 1\right)\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 2.35e-12)
     (* t_0 (+ x 1.0))
     (if (<= eps_m 1.55e+196)
       (+ 1.0 (* x (- (* x 0.25) 0.5)))
       (/ (* t_0 2.0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = t_0 * (x + 1.0);
	} else if (eps_m <= 1.55e+196) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = (t_0 * 2.0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 2.35d-12) then
        tmp = t_0 * (x + 1.0d0)
    else if (eps_m <= 1.55d+196) then
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    else
        tmp = (t_0 * 2.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = t_0 * (x + 1.0);
	} else if (eps_m <= 1.55e+196) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = (t_0 * 2.0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 2.35e-12:
		tmp = t_0 * (x + 1.0)
	elif eps_m <= 1.55e+196:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	else:
		tmp = (t_0 * 2.0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 2.35e-12)
		tmp = Float64(t_0 * Float64(x + 1.0));
	elseif (eps_m <= 1.55e+196)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	else
		tmp = Float64(Float64(t_0 * 2.0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 2.35e-12)
		tmp = t_0 * (x + 1.0);
	elseif (eps_m <= 1.55e+196)
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	else
		tmp = (t_0 * 2.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 2.35e-12], N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.55e+196], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 2.34999999999999988e-12

   1. Initial program 60.9%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around 0 34.7%

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

      1. associate-+r+ 73.3%

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

      2. mul-1-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-inverses 73.3%

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

      5. distribute-lft-out 73.3%

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

      6. distribute-rgt1-in 73.9%

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

      7. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in eps around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 2.34999999999999988e-12 < eps < 1.55000000000000005e196

   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 70.5%

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

   6. Taylor expanded in eps around inf 70.5%

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

      1. cancel-sign-sub-inv 70.5%

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

      2. metadata-eval 70.5%

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

      3. mul-1-neg 70.5%

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

      4. distribute-rgt-in 70.5%

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

      5. *-lft-identity 70.5%

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

      6. +-commutative 70.5%

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

      7. *-commutative 70.5%

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

      8. fma-undefine 70.5%

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

      9. *-lft-identity 70.5%

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

      10. fma-undefine 70.5%

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

      11. *-commutative 70.5%

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

      12. +-commutative 70.5%

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

      13. *-lft-identity 70.5%

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

      14. distribute-rgt-in 70.5%

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

      15. distribute-rgt-neg-in 70.5%

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

      16. distribute-neg-in 70.5%

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

      17. metadata-eval 70.5%

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

      18. unsub-neg 70.5%

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

   8. Simplified 70.5%

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

   9. Taylor expanded in eps around 0 57.8%

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

       1. mul-1-neg 57.8%

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

   11. Simplified 57.8%

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

   12. Taylor expanded in x around 0 64.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]

   if 1.55000000000000005e196 < 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 82.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 eps around 0 59.8%

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

      1. rec-exp 59.8%

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

      2. neg-mul-1 59.8%

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

      3. count-2 59.8%

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

      4. neg-mul-1 59.8%

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

   8. Simplified 59.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+196}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot 2}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 25500000:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 25500000.0)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ 1.0 (exp (* eps_m x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 25500000.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 25500000.0d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 25500000.0) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 25500000.0:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 25500000.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 25500000.0)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 25500000.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.55e7

   1. Initial program 62.4%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in eps around 0 35.8%

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

      1. associate-+r+ 73.8%

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

      2. mul-1-neg 73.8%

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

      3. sub-neg 73.8%

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

      4. +-inverses 73.8%

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

      5. distribute-lft-out 73.8%

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

      6. distribute-rgt1-in 74.3%

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

      7. mul-1-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in eps around 0 74.3%

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

      1. +-commutative 74.3%

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

   10. Simplified 74.3%

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

   if 2.55e7 < 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 87.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   4. 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 63.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 25500000:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.35e-12)
   (* (exp (- x)) (+ x 1.0))
   (+ 1.0 (* x (- (* x 0.25) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.35e-12) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	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 <= 2.35d-12) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    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 <= 2.35e-12) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 2.35e-12:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.35e-12)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 2.35e-12)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.35e-12], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.34999999999999988e-12

   1. Initial program 60.9%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around 0 34.7%

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

      1. associate-+r+ 73.3%

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

      2. mul-1-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-inverses 73.3%

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

      5. distribute-lft-out 73.3%

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

      6. distribute-rgt1-in 73.9%

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

      7. mul-1-neg 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in eps around 0 73.9%

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

      1. +-commutative 73.9%

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

   10. Simplified 73.9%

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

   if 2.34999999999999988e-12 < 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 99.9%

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

   5. Taylor expanded in x around 0 69.5%

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

   6. Taylor expanded in eps around inf 69.5%

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

      1. cancel-sign-sub-inv 69.5%

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

      2. metadata-eval 69.5%

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

      3. mul-1-neg 69.5%

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

      4. distribute-rgt-in 69.5%

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

      5. *-lft-identity 69.5%

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

      6. +-commutative 69.5%

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

      7. *-commutative 69.5%

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

      8. fma-undefine 69.5%

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

      9. *-lft-identity 69.5%

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

      10. fma-undefine 69.5%

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

      11. *-commutative 69.5%

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

      12. +-commutative 69.5%

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

      13. *-lft-identity 69.5%

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

      14. distribute-rgt-in 69.5%

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

      15. distribute-rgt-neg-in 69.5%

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

      16. distribute-neg-in 69.5%

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

      17. metadata-eval 69.5%

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

      18. unsub-neg 69.5%

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

   8. Simplified 69.5%

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

   9. Taylor expanded in eps around 0 58.3%

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

       1. mul-1-neg 58.3%

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

   11. Simplified 58.3%

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

   12. Taylor expanded in x around 0 60.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.35 \cdot 10^{-12}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.2)
   (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
   (* x (exp (- x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = x * exp(-x);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.2d0) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else
        tmp = x * exp(-x)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = x * Math.exp(-x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.2:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	else:
		tmp = x * math.exp(-x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	else
		tmp = Float64(x * exp(Float64(-x)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	else
		tmp = x * exp(-x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.2], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in eps around inf 79.0%

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

      1. cancel-sign-sub-inv 79.0%

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

      2. metadata-eval 79.0%

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

      3. mul-1-neg 79.0%

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

      4. distribute-rgt-in 79.0%

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

      5. *-lft-identity 79.0%

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

      6. +-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. fma-undefine 79.0%

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

      9. *-lft-identity 79.0%

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

      10. fma-undefine 79.0%

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

      11. *-commutative 79.0%

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

      12. +-commutative 79.0%

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

      13. *-lft-identity 79.0%

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

      14. distribute-rgt-in 79.0%

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

      15. distribute-rgt-neg-in 79.0%

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

      16. distribute-neg-in 79.0%

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

      17. metadata-eval 79.0%

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

      18. unsub-neg 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in eps around 0 77.2%

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

       1. mul-1-neg 77.2%

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

   11. Simplified 77.2%

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

   12. Taylor expanded in x around 0 72.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 2.2000000000000002 < x

   1. Initial program 98.6%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in eps around 0 61.2%

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

      1. associate-+r+ 62.6%

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

      2. mul-1-neg 62.6%

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

      3. sub-neg 62.6%

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

      4. +-inverses 62.6%

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

      5. distribute-lft-out 62.6%

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

      6. distribute-rgt1-in 62.6%

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

      7. mul-1-neg 62.6%

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

   7. Simplified 62.6%

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

   8. Taylor expanded in x around inf 61.5%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+154}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.8e+154)
   (+ 1.0 (* x (- (* x 0.25) 0.5)))
   (if (<= x 0.022) (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.8e+154) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else if (x <= 0.022) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2.8d+154)) then
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    else if (x <= 0.022d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.8e+154) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else if (x <= 0.022) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.8e+154:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	elif x <= 0.022:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.8e+154)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	elseif (x <= 0.022)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.8e+154)
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	elseif (x <= 0.022)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.8e+154], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.022], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.7999999999999999e154

   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 63.7%

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

   6. Taylor expanded in eps around inf 63.7%

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

      1. cancel-sign-sub-inv 63.7%

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

      2. metadata-eval 63.7%

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

      3. mul-1-neg 63.7%

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

      4. distribute-rgt-in 63.7%

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

      5. *-lft-identity 63.7%

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

      6. +-commutative 63.7%

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

      7. *-commutative 63.7%

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

      8. fma-undefine 63.7%

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

      9. *-lft-identity 63.7%

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

      10. fma-undefine 63.7%

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

      11. *-commutative 63.7%

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

      12. +-commutative 63.7%

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

      13. *-lft-identity 63.7%

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

      14. distribute-rgt-in 63.7%

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

      15. distribute-rgt-neg-in 63.7%

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

      16. distribute-neg-in 63.7%

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

      17. metadata-eval 63.7%

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

      18. unsub-neg 63.7%

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

   8. Simplified 63.7%

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

   9. Taylor expanded in eps around 0 100.0%

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

       1. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 100.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]

   if -2.7999999999999999e154 < x < 0.021999999999999999

   1. Initial program 58.4%

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

   3. Simplified 58.4%

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

   5. Taylor expanded in x around 0 42.3%

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

   6. Taylor expanded in eps around inf 80.9%

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

      1. cancel-sign-sub-inv 80.9%

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

      2. metadata-eval 80.9%

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

      3. mul-1-neg 80.9%

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

      4. distribute-rgt-in 80.9%

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

      5. *-lft-identity 80.9%

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

      6. +-commutative 80.9%

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

      7. *-commutative 80.9%

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

      8. fma-undefine 80.9%

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

      9. *-lft-identity 80.9%

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

      10. fma-undefine 80.9%

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

      11. *-commutative 80.9%

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

      12. +-commutative 80.9%

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

      13. *-lft-identity 80.9%

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

      14. distribute-rgt-in 80.9%

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

      15. distribute-rgt-neg-in 80.9%

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

      16. distribute-neg-in 80.9%

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

      17. metadata-eval 80.9%

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

      18. unsub-neg 80.9%

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

   8. Simplified 80.9%

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

   9. Taylor expanded in x around 0 68.0%

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

       1. mul-1-neg 68.0%

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

       2. distribute-rgt-neg-in 68.0%

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

       3. mul-1-neg 68.0%

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

       4. distribute-lft-in 68.0%

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

       5. metadata-eval 68.0%

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

       6. neg-mul-1 68.0%

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

       7. sub-neg 68.0%

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

   11. Simplified 68.0%

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

   if 0.021999999999999999 < x

   1. Initial program 98.7%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in eps around 0 60.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 60.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 60.4%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 60.4%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 60.4%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 60.4%

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

   7. Simplified 60.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{+154}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{elif}\;x \leq 0.022:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.6% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
   0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in eps around inf 79.0%

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

      1. cancel-sign-sub-inv 79.0%

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

      2. metadata-eval 79.0%

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

      3. mul-1-neg 79.0%

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

      4. distribute-rgt-in 79.0%

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

      5. *-lft-identity 79.0%

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

      6. +-commutative 79.0%

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

      7. *-commutative 79.0%

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

      8. fma-undefine 79.0%

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

      9. *-lft-identity 79.0%

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

      10. fma-undefine 79.0%

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

      11. *-commutative 79.0%

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

      12. +-commutative 79.0%

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

      13. *-lft-identity 79.0%

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

      14. distribute-rgt-in 79.0%

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

      15. distribute-rgt-neg-in 79.0%

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

      16. distribute-neg-in 79.0%

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

      17. metadata-eval 79.0%

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

      18. unsub-neg 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in eps around 0 77.2%

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

       1. mul-1-neg 77.2%

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

   11. Simplified 77.2%

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

   12. Taylor expanded in x around 0 72.1%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 2.5 < x

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in eps around 0 61.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 61.2%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 61.2%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 61.2%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 61.2%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 61.2%

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

   7. Simplified 61.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 500.0) (+ 1.0 (* x (- (* x 0.25) 0.5))) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 500.0d0) then
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 500.0:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 500.0)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 500.0)
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 500.0], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 62.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in x around 0 44.0%

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

   6. Taylor expanded in eps around inf 78.7%

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

      1. cancel-sign-sub-inv 78.7%

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

      2. metadata-eval 78.7%

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

      3. mul-1-neg 78.7%

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

      4. distribute-rgt-in 78.7%

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

      5. *-lft-identity 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. fma-undefine 78.7%

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

      9. *-lft-identity 78.7%

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

      10. fma-undefine 78.7%

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

      11. *-commutative 78.7%

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

      12. +-commutative 78.7%

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

      13. *-lft-identity 78.7%

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

      14. distribute-rgt-in 78.7%

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

      15. distribute-rgt-neg-in 78.7%

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

      16. distribute-neg-in 78.7%

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

      17. metadata-eval 78.7%

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

      18. unsub-neg 78.7%

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

   8. Simplified 78.7%

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

   9. Taylor expanded in eps around 0 76.5%

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

       1. mul-1-neg 76.5%

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

   11. Simplified 76.5%

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

   12. Taylor expanded in x around 0 68.8%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]

   if 500 < 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%

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

   5. Taylor expanded in eps around 0 62.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 62.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 62.9%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 62.9%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 62.9%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 62.9%

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

   7. Simplified 62.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 490:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 490.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 490.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 490.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 490.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 490.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 490.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 490

   1. Initial program 62.1%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around inf 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around 0 60.7%

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

       1. mul-1-neg 60.7%

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

       2. unsub-neg 60.7%

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

   11. Simplified 60.7%

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

   12. Taylor expanded in x around 0 61.2%

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

   if 490 < 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%

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

   5. Taylor expanded in eps around 0 62.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 62.9%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

      2. mul-1-neg 62.9%

         \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      3. rec-exp 62.9%

         \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

      4. +-inverses 62.9%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 62.9%

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

   7. Simplified 62.9%

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

Alternative 15: 16.7% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.3%

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

3. Simplified 64.7%

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

5. Taylor expanded in eps around 0 18.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
6. Step-by-step derivation

   1. div-sub 18.7%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]

   2. mul-1-neg 18.7%

      \[\leadsto 0.5 \cdot \left(\frac{e^{\color{blue}{-x}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

   3. rec-exp 18.6%

      \[\leadsto 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]

   4. +-inverses 18.8%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   5. metadata-eval 18.8%

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

7. Simplified 18.8%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))