NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 99.4%

Time: 18.4s

Alternatives: 16

Speedup: 2.0×

• 38.3% 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: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{t\_0 + \frac{1}{t\_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 1.75) (/ (+ t_0 (/ 1.0 t_0)) 2.0) (/ t_0 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 1.75) {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	} else {
		tmp = t_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 * (eps_m + (-1.0d0))))
    if (x <= 1.75d0) then
        tmp = (t_0 + (1.0d0 / t_0)) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 1.75) {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 1.75:
		tmp = (t_0 + (1.0 / t_0)) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(Float64(t_0 + Float64(1.0 / t_0)) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 1.75)
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	else
		tmp = t_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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.75], N[(N[(t$95$0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 59.4%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. add-sqr-sqrt 37.3%

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

      3. sqrt-unprod 95.5%

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

      4. sqr-neg 95.5%

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

      5. sqrt-unprod 61.4%

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

      6. add-sqr-sqrt 99.2%

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

      7. neg-mul-1 99.2%

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

      8. distribute-rgt-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. sub-neg 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

   7. Applied egg-rr 99.2%

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

   if 1.75 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

   7. Taylor expanded in x around inf 76.8%

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

4. Final simplification 92.9%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{t\_0 + \frac{1}{{e}^{\left(x \cdot eps\_m\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 4.7) (/ (+ t_0 (/ 1.0 (pow E (* x eps_m)))) 2.0) (/ t_0 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + (1.0 / pow(((double) M_E), (x * eps_m)))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + (1.0 / Math.pow(Math.E, (x * eps_m)))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 4.7:
		tmp = (t_0 + (1.0 / math.pow(math.e, (x * eps_m)))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 4.7)
		tmp = Float64(Float64(t_0 + Float64(1.0 / (exp(1) ^ Float64(x * eps_m)))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 4.7)
		tmp = (t_0 + (1.0 / (2.71828182845904523536 ^ (x * eps_m)))) / 2.0;
	else
		tmp = t_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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.7], N[(N[(t$95$0 + N[(1.0 / N[Power[E, N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.70000000000000018

   1. Initial program 59.4%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

   8. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. exp-prod 98.8%

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

      4. *-commutative 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. exp-1-e 98.8%

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

       2. *-commutative 98.8%

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

   12. Simplified 98.8%

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

   if 4.70000000000000018 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

   7. Taylor expanded in x around inf 76.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{t\_0 + \frac{1}{e^{x \cdot eps\_m}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 4.7) (/ (+ t_0 (/ 1.0 (exp (* x eps_m)))) 2.0) (/ t_0 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + (1.0 / exp((x * eps_m)))) / 2.0;
	} else {
		tmp = t_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 * (eps_m + (-1.0d0))))
    if (x <= 4.7d0) then
        tmp = (t_0 + (1.0d0 / exp((x * eps_m)))) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + (1.0 / Math.exp((x * eps_m)))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 4.7:
		tmp = (t_0 + (1.0 / math.exp((x * eps_m)))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 4.7)
		tmp = Float64(Float64(t_0 + Float64(1.0 / exp(Float64(x * eps_m)))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 4.7)
		tmp = (t_0 + (1.0 / exp((x * eps_m)))) / 2.0;
	else
		tmp = t_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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.7], N[(N[(t$95$0 + N[(1.0 / N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.70000000000000018

   1. Initial program 59.4%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

   8. Simplified 98.8%

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

   if 4.70000000000000018 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

   7. Taylor expanded in x around inf 76.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{t\_0 + {e}^{\left(eps\_m \cdot \left(-x\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 4.7) (/ (+ t_0 (pow E (* eps_m (- x)))) 2.0) (/ t_0 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + pow(((double) M_E), (eps_m * -x))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + Math.pow(Math.E, (eps_m * -x))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 4.7:
		tmp = (t_0 + math.pow(math.e, (eps_m * -x))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 4.7)
		tmp = Float64(Float64(t_0 + (exp(1) ^ Float64(eps_m * Float64(-x)))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 4.7)
		tmp = (t_0 + (2.71828182845904523536 ^ (eps_m * -x))) / 2.0;
	else
		tmp = t_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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.7], N[(N[(t$95$0 + N[Power[E, N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.70000000000000018

   1. Initial program 59.4%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

   8. Simplified 98.8%

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

      1. *-commutative 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. exp-prod 98.8%

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

      4. *-commutative 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. exp-1-e 98.8%

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

       2. *-commutative 98.8%

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

   12. Simplified 98.8%

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

       1. inv-pow 98.8%

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

       2. pow-pow 98.8%

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

   14. Applied egg-rr 98.8%

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

       1. *-commutative 98.8%

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

       2. mul-1-neg 98.8%

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

       3. *-commutative 98.8%

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

       4. distribute-rgt-neg-in 98.8%

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

   16. Simplified 98.8%

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

   if 4.70000000000000018 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

   7. Taylor expanded in x around inf 76.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + {e}^{\left(\varepsilon \cdot \left(-x\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{t\_0 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 4.7) (/ (+ t_0 (exp (* eps_m (- x)))) 2.0) (/ t_0 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + exp((eps_m * -x))) / 2.0;
	} else {
		tmp = t_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 * (eps_m + (-1.0d0))))
    if (x <= 4.7d0) then
        tmp = (t_0 + exp((eps_m * -x))) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 4.7) {
		tmp = (t_0 + Math.exp((eps_m * -x))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 4.7:
		tmp = (t_0 + math.exp((eps_m * -x))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 4.7)
		tmp = Float64(Float64(t_0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 4.7)
		tmp = (t_0 + exp((eps_m * -x))) / 2.0;
	else
		tmp = t_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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.7], N[(N[(t$95$0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.70000000000000018

   1. Initial program 59.4%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

   8. Simplified 98.8%

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

      1. rec-exp 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

   10. Applied egg-rr 98.8%

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

   if 4.70000000000000018 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

   7. Taylor expanded in x around inf 76.8%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Simplified 65.9%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Final simplification 99.1%

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

Alternative 7: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq -3.3 \cdot 10^{-249}:\\ \;\;\;\;\frac{1 + \frac{1}{{e}^{\left(x \cdot eps\_m\right)}}}{2}\\ \mathbf{elif}\;x \leq 8:\\ \;\;\;\;\frac{t\_0 + \frac{1}{1 + x \cdot \left(eps\_m + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -3.3e-249)
     (/ (+ 1.0 (/ 1.0 (pow E (* x eps_m)))) 2.0)
     (if (<= x 8.0)
       (/ (+ t_0 (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0))))) 2.0)
       (/ t_0 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -3.3e-249) {
		tmp = (1.0 + (1.0 / pow(((double) M_E), (x * eps_m)))) / 2.0;
	} else if (x <= 8.0) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -3.3e-249) {
		tmp = (1.0 + (1.0 / Math.pow(Math.E, (x * eps_m)))) / 2.0;
	} else if (x <= 8.0) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -3.3e-249:
		tmp = (1.0 + (1.0 / math.pow(math.e, (x * eps_m)))) / 2.0
	elif x <= 8.0:
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -3.3e-249)
		tmp = Float64(Float64(1.0 + Float64(1.0 / (exp(1) ^ Float64(x * eps_m)))) / 2.0);
	elseif (x <= 8.0)
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -3.3e-249)
		tmp = (1.0 + (1.0 / (2.71828182845904523536 ^ (x * eps_m)))) / 2.0;
	elseif (x <= 8.0)
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	else
		tmp = t_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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.3e-249], N[(N[(1.0 + N[(1.0 / N[Power[E, N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.0], N[(N[(t$95$0 + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e-249

   1. Initial program 67.7%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in eps around inf 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 inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. exp-prod 100.0%

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

      4. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. exp-1-e 100.0%

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

       2. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 72.0%

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

   if -3.3e-249 < x < 8

   1. Initial program 49.8%

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

   3. Simplified 43.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}{2}} \]
   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 x around 0 84.5%

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

   if 8 < 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 inf 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

   7. Taylor expanded in x around inf 76.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-249}:\\ \;\;\;\;\frac{1 + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2}\\ \mathbf{elif}\;x \leq 8:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 124:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+91} \lor \neg \left(x \leq 3.1 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 124.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (or (<= x 1.55e+91) (not (<= x 3.1e+130)))
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
     (/ (+ 1.0 (exp x)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 124.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if ((x <= 1.55e+91) || !(x <= 3.1e+130)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 124.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if ((x <= 1.55d+91) .or. (.not. (x <= 3.1d+130))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 124.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if ((x <= 1.55e+91) || !(x <= 3.1e+130)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 124.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif (x <= 1.55e+91) or not (x <= 3.1e+130):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 124.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif ((x <= 1.55e+91) || !(x <= 3.1e+130))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 124.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif ((x <= 1.55e+91) || ~((x <= 3.1e+130)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 124.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.55e+91], N[Not[LessEqual[x, 3.1e+130]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 124

   1. Initial program 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in x around 0 41.3%

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

   6. Taylor expanded in x around 0 44.2%

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

   7. Taylor expanded in eps around inf 66.0%

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

      1. associate-*r* 66.0%

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

      2. neg-mul-1 66.0%

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

      3. *-commutative 66.0%

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

   9. Simplified 66.0%

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

   if 124 < x < 1.54999999999999999e91 or 3.1e130 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.3%

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

   6. Taylor expanded in x around 0 62.2%

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

   if 1.54999999999999999e91 < x < 3.1e130

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 92.5%

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

      1. *-commutative 92.5%

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

   8. Simplified 92.5%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

       1. *-un-lft-identity 3.1%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 92.4%

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

       4. sqr-neg 92.4%

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

       5. sqrt-unprod 92.4%

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

       6. add-sqr-sqrt 92.4%

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

   13. Applied egg-rr 92.4%

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

       1. *-lft-identity 92.4%

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

   15. Simplified 92.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 124:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+91} \lor \neg \left(x \leq 3.1 \cdot 10^{+130}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+91} \lor \neg \left(x \leq 2.8 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 360.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 2.2e+91) (not (<= x 2.8e+133)))
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
     (/ (+ 1.0 (exp x)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 2.2e+91) || !(x <= 2.8e+133)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 360.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 2.2d+91) .or. (.not. (x <= 2.8d+133))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 360.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 2.2e+91) || !(x <= 2.8e+133)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 360.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 2.2e+91) or not (x <= 2.8e+133):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 360.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 2.2e+91) || !(x <= 2.8e+133))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 360.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 2.2e+91) || ~((x <= 2.8e+133)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 360.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.2e+91], N[Not[LessEqual[x, 2.8e+133]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 360

   1. Initial program 59.4%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.8%

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

      1. *-commutative 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in eps around 0 79.9%

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

       1. mul-1-neg 79.9%

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

   11. Simplified 79.9%

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

   if 360 < x < 2.19999999999999999e91 or 2.80000000000000016e133 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 22.3%

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

   6. Taylor expanded in x around 0 62.2%

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

   if 2.19999999999999999e91 < x < 2.80000000000000016e133

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 92.5%

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

      1. *-commutative 92.5%

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

   8. Simplified 92.5%

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

   9. Taylor expanded in eps around 0 3.1%

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

       1. mul-1-neg 3.1%

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

   11. Simplified 3.1%

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

       1. *-un-lft-identity 3.1%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 92.4%

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

       4. sqr-neg 92.4%

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

       5. sqrt-unprod 92.4%

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

       6. add-sqr-sqrt 92.4%

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

   13. Applied egg-rr 92.4%

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

       1. *-lft-identity 92.4%

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

   15. Simplified 92.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+91} \lor \neg \left(x \leq 2.8 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-52}:\\ \;\;\;\;\frac{1 + \frac{1}{{e}^{\left(x \cdot eps\_m\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1e-52)
   (/ (+ 1.0 (/ 1.0 (pow E (* x eps_m)))) 2.0)
   (/ (exp (* x (+ eps_m -1.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1e-52) {
		tmp = (1.0 + (1.0 / pow(((double) M_E), (x * eps_m)))) / 2.0;
	} else {
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1e-52) {
		tmp = (1.0 + (1.0 / Math.pow(Math.E, (x * eps_m)))) / 2.0;
	} else {
		tmp = Math.exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1e-52:
		tmp = (1.0 + (1.0 / math.pow(math.e, (x * eps_m)))) / 2.0
	else:
		tmp = math.exp((x * (eps_m + -1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1e-52)
		tmp = Float64(Float64(1.0 + Float64(1.0 / (exp(1) ^ Float64(x * eps_m)))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps_m + -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1e-52)
		tmp = (1.0 + (1.0 / (2.71828182845904523536 ^ (x * eps_m)))) / 2.0;
	else
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1e-52], N[(N[(1.0 + N[(1.0 / N[Power[E, N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-52

   1. Initial program 59.5%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in eps around inf 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 inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. exp-prod 100.0%

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

      4. *-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. exp-1-e 100.0%

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

       2. *-commutative 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 81.8%

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

   if 1e-52 < x

   1. Initial program 93.2%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in eps around inf 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 x around 0 31.5%

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

   7. Taylor expanded in x around inf 68.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-52}:\\ \;\;\;\;\frac{1 + \frac{1}{{e}^{\left(x \cdot \varepsilon\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x \cdot eps\_m}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9.8e-53)
   (/ (+ 1.0 (/ 1.0 (exp (* x eps_m)))) 2.0)
   (/ (exp (* x (+ eps_m -1.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.8e-53) {
		tmp = (1.0 + (1.0 / exp((x * eps_m)))) / 2.0;
	} else {
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 9.8d-53) then
        tmp = (1.0d0 + (1.0d0 / exp((x * eps_m)))) / 2.0d0
    else
        tmp = exp((x * (eps_m + (-1.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.8e-53) {
		tmp = (1.0 + (1.0 / Math.exp((x * eps_m)))) / 2.0;
	} else {
		tmp = Math.exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9.8e-53:
		tmp = (1.0 + (1.0 / math.exp((x * eps_m)))) / 2.0
	else:
		tmp = math.exp((x * (eps_m + -1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9.8e-53)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(Float64(x * eps_m)))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps_m + -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 9.8e-53)
		tmp = (1.0 + (1.0 / exp((x * eps_m)))) / 2.0;
	else
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9.8e-53], N[(N[(1.0 + N[(1.0 / N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.79999999999999926e-53

   1. Initial program 59.5%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in eps around inf 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 inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 81.8%

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

   if 9.79999999999999926e-53 < x

   1. Initial program 93.2%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in eps around inf 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 x around 0 31.5%

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

   7. Taylor expanded in x around inf 68.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 85.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9.2e-54)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (exp (* x (+ eps_m -1.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.2e-54) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 9.2d-54) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = exp((x * (eps_m + (-1.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.2e-54) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = Math.exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9.2e-54:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = math.exp((x * (eps_m + -1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9.2e-54)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps_m + -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 9.2e-54)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9.2e-54], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.1999999999999996e-54

   1. Initial program 59.5%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in eps around inf 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 inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around 0 83.5%

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

       1. mul-1-neg 83.5%

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

   11. Simplified 83.5%

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

   if 9.1999999999999996e-54 < x

   1. Initial program 93.2%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in eps around inf 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 x around 0 31.5%

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

   7. Taylor expanded in x around inf 68.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 152:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+86} \lor \neg \left(x \leq 5.2 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 152.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (or (<= x 1.8e+86) (not (<= x 5.2e+131)))
     (/ (+ (+ 1.0 (/ 1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
     (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 152.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if ((x <= 1.8e+86) || !(x <= 5.2e+131)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 152.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if ((x <= 1.8d+86) .or. (.not. (x <= 5.2d+131))) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 152.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if ((x <= 1.8e+86) || !(x <= 5.2e+131)) {
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 152.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif (x <= 1.8e+86) or not (x <= 5.2e+131):
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 152.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif ((x <= 1.8e+86) || !(x <= 5.2e+131))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 152.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif ((x <= 1.8e+86) || ~((x <= 5.2e+131)))
		tmp = ((1.0 + (1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 152.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.8e+86], N[Not[LessEqual[x, 5.2e+131]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 152

   1. Initial program 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in x around 0 41.3%

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

   6. Taylor expanded in x around 0 44.2%

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

   7. Taylor expanded in eps around inf 66.0%

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

      1. associate-*r* 66.0%

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

      2. neg-mul-1 66.0%

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

      3. *-commutative 66.0%

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

   9. Simplified 66.0%

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

   if 152 < x < 1.80000000000000003e86 or 5.2e131 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 23.7%

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

   6. Taylor expanded in x around 0 63.0%

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

   if 1.80000000000000003e86 < x < 5.2e131

   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 49.2%

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

   6. Taylor expanded in eps around inf 32.2%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 152:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+86} \lor \neg \left(x \leq 5.2 \cdot 10^{+131}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.3% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 7.6) (/ (- 2.0 (* x eps_m)) 2.0) (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 7.6) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 7.6d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 7.6) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 7.6:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 7.6)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 7.6)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 7.6], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999996

   1. Initial program 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in x around 0 41.3%

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

   6. Taylor expanded in x around 0 44.2%

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

   7. Taylor expanded in eps around inf 66.0%

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

      1. associate-*r* 66.0%

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

      2. neg-mul-1 66.0%

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

      3. *-commutative 66.0%

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

   9. Simplified 66.0%

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

   if 7.5999999999999996 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 28.1%

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

   6. Taylor expanded in eps around inf 16.5%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.6% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 24.5) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 24.5d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 24.5) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 24.5:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 24.5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 24.5)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 24.5], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 24.5

   1. Initial program 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in x around 0 62.4%

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

   if 24.5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 28.1%

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

   6. Taylor expanded in eps around inf 16.5%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 24.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.5% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Simplified 70.8%

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

5. Taylor expanded in x around 0 45.7%

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

6. Final simplification 45.7%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))