NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.9%

Time: 22.6s

Alternatives: 18

Speedup: 1.8×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.002:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.002)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.002) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.002d0) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.002) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.002:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.002)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.002)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.002], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2e-3

   1. Initial program 64.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in eps around 0 33.1%

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

   7. Simplified 69.7%

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

   if 2e-3 < eps

   1. Initial program 100.0%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in eps around inf 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 -inf 100.0%

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

      1. rec-exp 100.0%

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

      2. distribute-rgt-out-- 100.0%

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

      3. sub-neg 100.0%

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

      4. metadata-eval 100.0%

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

      5. +-commutative 100.0%

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

      6. remove-double-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. cancel-sign-sub-inv 100.0%

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

      11. metadata-eval 100.0%

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

      12. *-lft-identity 100.0%

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

      13. distribute-neg-in 100.0%

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

      14. metadata-eval 100.0%

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

      15. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.002:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% 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 73.5%

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

3. Simplified 69.2%

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

5. Taylor expanded in eps around inf 98.4%

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

6. Final simplification 98.4%

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

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - 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)))) + exp((x * ((-1.0d0) - 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))) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.exp((x * (eps_m + -1.0))) + math.exp((x * (-1.0 - 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))) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - 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[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Simplified 69.2%

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

5. Taylor expanded in eps around inf 98.4%

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

6. Taylor expanded in x around -inf 98.4%

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

   1. rec-exp 98.4%

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

   2. distribute-rgt-out-- 98.4%

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

   3. sub-neg 98.4%

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

   4. metadata-eval 98.4%

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

   5. +-commutative 98.4%

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

   6. remove-double-neg 98.4%

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

   7. mul-1-neg 98.4%

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

   8. sub-neg 98.4%

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

   9. distribute-rgt-neg-in 98.4%

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

   10. cancel-sign-sub-inv 98.4%

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

   11. metadata-eval 98.4%

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

   12. *-lft-identity 98.4%

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

   13. distribute-neg-in 98.4%

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

   14. metadata-eval 98.4%

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

   15. unsub-neg 98.4%

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 4: 87.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 10^{+40}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \mathbf{elif}\;eps\_m \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) - \left(-1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;eps\_m \leq 1.35 \cdot 10^{+245}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\left(\frac{1}{eps\_m} + \frac{t\_0}{eps\_m}\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= eps_m 1.0)
     (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
     (if (<= eps_m 1e+40)
       (/ (+ 1.0 t_0) 2.0)
       (if (<= eps_m 1.55e+63)
         (/
          (-
           (+ 1.0 (/ 1.0 eps_m))
           (* (+ -1.0 (/ 1.0 eps_m)) (exp (* x (- -1.0 eps_m)))))
          2.0)
         (if (<= eps_m 1.35e+245)
           (/ (* eps_m (- (+ (/ 1.0 eps_m) (/ t_0 eps_m)) x)) 2.0)
           (/
            (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (pow (+ eps_m -1.0) 2.0))))))
            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 (eps_m <= 1.0) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 1e+40) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (eps_m <= 1.55e+63) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 + (1.0 / eps_m)) * exp((x * (-1.0 - eps_m))))) / 2.0;
	} else if (eps_m <= 1.35e+245) {
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * pow((eps_m + -1.0), 2.0)))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (eps_m <= 1.0d0) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else if (eps_m <= 1d+40) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if (eps_m <= 1.55d+63) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - (((-1.0d0) + (1.0d0 / eps_m)) * exp((x * ((-1.0d0) - eps_m))))) / 2.0d0
    else if (eps_m <= 1.35d+245) then
        tmp = (eps_m * (((1.0d0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 1e+40) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (eps_m <= 1.55e+63) {
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 + (1.0 / eps_m)) * Math.exp((x * (-1.0 - eps_m))))) / 2.0;
	} else if (eps_m <= 1.35e+245) {
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * Math.pow((eps_m + -1.0), 2.0)))))) / 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 eps_m <= 1.0:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	elif eps_m <= 1e+40:
		tmp = (1.0 + t_0) / 2.0
	elif eps_m <= 1.55e+63:
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 + (1.0 / eps_m)) * math.exp((x * (-1.0 - eps_m))))) / 2.0
	elif eps_m <= 1.35e+245:
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * math.pow((eps_m + -1.0), 2.0)))))) / 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 (eps_m <= 1.0)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	elseif (eps_m <= 1e+40)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (eps_m <= 1.55e+63)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(Float64(-1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(-1.0 - eps_m))))) / 2.0);
	elseif (eps_m <= 1.35e+245)
		tmp = Float64(Float64(eps_m * Float64(Float64(Float64(1.0 / eps_m) + Float64(t_0 / eps_m)) - x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	elseif (eps_m <= 1e+40)
		tmp = (1.0 + t_0) / 2.0;
	elseif (eps_m <= 1.55e+63)
		tmp = ((1.0 + (1.0 / eps_m)) - ((-1.0 + (1.0 / eps_m)) * exp((x * (-1.0 - eps_m))))) / 2.0;
	elseif (eps_m <= 1.35e+245)
		tmp = (eps_m * (((1.0 / eps_m) + (t_0 / eps_m)) - x)) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * ((eps_m + -1.0) ^ 2.0)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 1.0], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1e+40], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.55e+63], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.35e+245], N[(N[(eps$95$m * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(t$95$0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < 1

   1. Initial program 64.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in eps around 0 33.1%

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

   7. Simplified 69.7%

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

   if 1 < eps < 1.00000000000000003e40

   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 0 100.0%

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

   if 1.00000000000000003e40 < eps < 1.55e63

   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 0 100.0%

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

   if 1.55e63 < eps < 1.34999999999999996e245

   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 72.1%

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

   6. Taylor expanded in eps around inf 84.6%

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

   7. Taylor expanded in eps around -inf 84.6%

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

      1. associate-*r* 84.6%

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

      2. neg-mul-1 84.6%

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

      3. sub-neg 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-*l* 84.6%

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

      6. sub-neg 84.6%

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

      7. distribute-lft-in 84.6%

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

      8. metadata-eval 84.6%

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

      9. mul-1-neg 84.6%

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

      10. remove-double-neg 84.6%

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

   9. Simplified 84.6%

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

   if 1.34999999999999996e245 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in eps around inf 74.1%

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

   7. Taylor expanded in eps around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-rgt-neg-in 74.1%

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

      3. mul-1-neg 74.1%

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

      4. sub-neg 74.1%

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

      5. distribute-lft-in 74.1%

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

      6. metadata-eval 74.1%

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

      7. mul-1-neg 74.1%

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

      8. remove-double-neg 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in x around 0 94.6%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right) - 1\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 10^{+40}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+63}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{+245}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(\frac{1}{\varepsilon} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 1.35 \cdot 10^{+245}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\left(\frac{1}{eps\_m} + \frac{e^{x \cdot \left(eps\_m + -1\right)}}{eps\_m}\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (if (<= eps_m 1.35e+245)
     (/
      (* eps_m (- (+ (/ 1.0 eps_m) (/ (exp (* x (+ eps_m -1.0))) eps_m)) x))
      2.0)
     (/ (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (pow (+ eps_m -1.0) 2.0)))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 1.35e+245) {
		tmp = (eps_m * (((1.0 / eps_m) + (exp((x * (eps_m + -1.0))) / eps_m)) - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * pow((eps_m + -1.0), 2.0)))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.0d0) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else if (eps_m <= 1.35d+245) then
        tmp = (eps_m * (((1.0d0 / eps_m) + (exp((x * (eps_m + (-1.0d0)))) / eps_m)) - x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 1.35e+245) {
		tmp = (eps_m * (((1.0 / eps_m) + (Math.exp((x * (eps_m + -1.0))) / eps_m)) - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * Math.pow((eps_m + -1.0), 2.0)))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	elif eps_m <= 1.35e+245:
		tmp = (eps_m * (((1.0 / eps_m) + (math.exp((x * (eps_m + -1.0))) / eps_m)) - x)) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * math.pow((eps_m + -1.0), 2.0)))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	elseif (eps_m <= 1.35e+245)
		tmp = Float64(Float64(eps_m * Float64(Float64(Float64(1.0 / eps_m) + Float64(exp(Float64(x * Float64(eps_m + -1.0))) / eps_m)) - x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	elseif (eps_m <= 1.35e+245)
		tmp = (eps_m * (((1.0 / eps_m) + (exp((x * (eps_m + -1.0))) / eps_m)) - x)) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * ((eps_m + -1.0) ^ 2.0)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.35e+245], N[(N[(eps$95$m * N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 64.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in eps around 0 33.1%

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

   7. Simplified 69.7%

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

   if 1 < eps < 1.34999999999999996e245

   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 69.4%

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

   6. Taylor expanded in eps around inf 79.6%

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

   7. Taylor expanded in eps around -inf 79.6%

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

      1. associate-*r* 79.6%

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

      2. neg-mul-1 79.6%

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

      3. sub-neg 79.6%

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

      4. *-commutative 79.6%

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

      5. associate-*l* 79.6%

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

      6. sub-neg 79.6%

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

      7. distribute-lft-in 79.6%

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

      8. metadata-eval 79.6%

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

      9. mul-1-neg 79.6%

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

      10. remove-double-neg 79.6%

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

   9. Simplified 79.6%

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

   if 1.34999999999999996e245 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in eps around inf 74.1%

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

   7. Taylor expanded in eps around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-rgt-neg-in 74.1%

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

      3. mul-1-neg 74.1%

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

      4. sub-neg 74.1%

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

      5. distribute-lft-in 74.1%

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

      6. metadata-eval 74.1%

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

      7. mul-1-neg 74.1%

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

      8. remove-double-neg 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in x around 0 94.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{+245}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(\frac{1}{\varepsilon} + \frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon}\right) - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 1.2 \cdot 10^{+245}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{eps\_m} - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (if (<= eps_m 1.2e+245)
     (/ (* eps_m (- (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) eps_m) x)) 2.0)
     (/ (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (pow (+ eps_m -1.0) 2.0)))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 1.2e+245) {
		tmp = (eps_m * (((1.0 + exp((x * (eps_m + -1.0)))) / eps_m) - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * pow((eps_m + -1.0), 2.0)))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.0d0) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else if (eps_m <= 1.2d+245) then
        tmp = (eps_m * (((1.0d0 + exp((x * (eps_m + (-1.0d0))))) / eps_m) - x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.0) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 1.2e+245) {
		tmp = (eps_m * (((1.0 + Math.exp((x * (eps_m + -1.0)))) / eps_m) - x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * Math.pow((eps_m + -1.0), 2.0)))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.0:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	elif eps_m <= 1.2e+245:
		tmp = (eps_m * (((1.0 + math.exp((x * (eps_m + -1.0)))) / eps_m) - x)) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * math.pow((eps_m + -1.0), 2.0)))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.0)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	elseif (eps_m <= 1.2e+245)
		tmp = Float64(Float64(eps_m * Float64(Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / eps_m) - x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.0)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	elseif (eps_m <= 1.2e+245)
		tmp = (eps_m * (((1.0 + exp((x * (eps_m + -1.0)))) / eps_m) - x)) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * ((eps_m + -1.0) ^ 2.0)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.2e+245], N[(N[(eps$95$m * N[(N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 64.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in eps around 0 33.1%

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

   7. Simplified 69.7%

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

   if 1 < eps < 1.1999999999999999e245

   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 69.4%

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

   6. Taylor expanded in eps around inf 79.6%

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

   7. Taylor expanded in eps around inf 79.6%

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

      1. mul-1-neg 79.6%

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

      2. distribute-rgt-neg-in 79.6%

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

      3. mul-1-neg 79.6%

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

      4. sub-neg 79.6%

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

      5. distribute-lft-in 79.6%

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

      6. metadata-eval 79.6%

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

      7. mul-1-neg 79.6%

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

      8. remove-double-neg 79.6%

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

   9. Simplified 79.6%

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

   if 1.1999999999999999e245 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 40.8%

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

   6. Taylor expanded in eps around inf 74.1%

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

   7. Taylor expanded in eps around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-rgt-neg-in 74.1%

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

      3. mul-1-neg 74.1%

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

      4. sub-neg 74.1%

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

      5. distribute-lft-in 74.1%

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

      6. metadata-eval 74.1%

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

      7. mul-1-neg 74.1%

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

      8. remove-double-neg 74.1%

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

   9. Simplified 74.1%

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

   10. Taylor expanded in x around 0 94.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.2 \cdot 10^{+245}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq -410:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{t\_0}{eps\_m} - x\right)}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* x (+ eps_m -1.0))))))
   (if (<= x -410.0)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 1.85e+158)
       (/ (* eps_m (- (/ t_0 eps_m) x)) 2.0)
       (if (<= x 9.2e+253) 0.0 (/ t_0 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -410.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.85e+158) {
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0;
	} else if (x <= 9.2e+253) {
		tmp = 0.0;
	} 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 = 1.0d0 + exp((x * (eps_m + (-1.0d0))))
    if (x <= (-410.0d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.85d+158) then
        tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0d0
    else if (x <= 9.2d+253) then
        tmp = 0.0d0
    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 = 1.0 + Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -410.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.85e+158) {
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0;
	} else if (x <= 9.2e+253) {
		tmp = 0.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -410.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.85e+158:
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0
	elif x <= 9.2e+253:
		tmp = 0.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0))))
	tmp = 0.0
	if (x <= -410.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.85e+158)
		tmp = Float64(Float64(eps_m * Float64(Float64(t_0 / eps_m) - x)) / 2.0);
	elseif (x <= 9.2e+253)
		tmp = 0.0;
	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 = 1.0 + exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -410.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.85e+158)
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0;
	elseif (x <= 9.2e+253)
		tmp = 0.0;
	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[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -410.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.85e+158], N[(N[(eps$95$m * N[(N[(t$95$0 / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.2e+253], 0.0, N[(t$95$0 / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -410

   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 59.1%

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

   6. Taylor expanded in eps around inf 85.4%

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

   7. Taylor expanded in eps around 0 100.0%

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

      1. neg-mul-1 100.0%

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

   9. Simplified 100.0%

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

   if -410 < x < 1.85000000000000005e158

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in x around 0 35.1%

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

   6. Taylor expanded in eps around inf 72.1%

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

   7. Taylor expanded in eps around inf 72.1%

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

      1. mul-1-neg 72.1%

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

      2. distribute-rgt-neg-in 72.1%

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

      3. mul-1-neg 72.1%

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

      4. sub-neg 72.1%

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

      5. distribute-lft-in 72.1%

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

      6. metadata-eval 72.1%

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

      7. mul-1-neg 72.1%

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

      8. remove-double-neg 72.1%

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

   9. Simplified 72.1%

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

   if 1.85000000000000005e158 < x < 9.1999999999999999e253

   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%

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

   5. Taylor expanded in eps around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. mul-1-neg 65.8%

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

      3. rec-exp 65.8%

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

      4. sub-neg 65.8%

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

      5. div-sub 65.8%

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

      6. mul-1-neg 65.8%

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

      7. rec-exp 65.8%

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

      8. +-inverses 65.8%

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

   7. Simplified 65.8%

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

   if 9.1999999999999999e253 < 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 47.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -410:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+158}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} - x\right)}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq -2.15 \cdot 10^{-232}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(eps\_m + -1\right)}^{2}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{t\_0}{eps\_m} - x\right)}{2}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+255}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* x (+ eps_m -1.0))))))
   (if (<= x -2.15e-232)
     (/ (+ 2.0 (* x (+ -1.0 (* 0.5 (* x (pow (+ eps_m -1.0) 2.0)))))) 2.0)
     (if (<= x 5.2e+157)
       (/ (* eps_m (- (/ t_0 eps_m) x)) 2.0)
       (if (<= x 5.7e+255) 0.0 (/ t_0 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -2.15e-232) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * pow((eps_m + -1.0), 2.0)))))) / 2.0;
	} else if (x <= 5.2e+157) {
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0;
	} else if (x <= 5.7e+255) {
		tmp = 0.0;
	} 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 = 1.0d0 + exp((x * (eps_m + (-1.0d0))))
    if (x <= (-2.15d-232)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (0.5d0 * (x * ((eps_m + (-1.0d0)) ** 2.0d0)))))) / 2.0d0
    else if (x <= 5.2d+157) then
        tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0d0
    else if (x <= 5.7d+255) then
        tmp = 0.0d0
    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 = 1.0 + Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -2.15e-232) {
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * Math.pow((eps_m + -1.0), 2.0)))))) / 2.0;
	} else if (x <= 5.2e+157) {
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0;
	} else if (x <= 5.7e+255) {
		tmp = 0.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -2.15e-232:
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * math.pow((eps_m + -1.0), 2.0)))))) / 2.0
	elif x <= 5.2e+157:
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0
	elif x <= 5.7e+255:
		tmp = 0.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0))))
	tmp = 0.0
	if (x <= -2.15e-232)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(0.5 * Float64(x * (Float64(eps_m + -1.0) ^ 2.0)))))) / 2.0);
	elseif (x <= 5.2e+157)
		tmp = Float64(Float64(eps_m * Float64(Float64(t_0 / eps_m) - x)) / 2.0);
	elseif (x <= 5.7e+255)
		tmp = 0.0;
	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 = 1.0 + exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -2.15e-232)
		tmp = (2.0 + (x * (-1.0 + (0.5 * (x * ((eps_m + -1.0) ^ 2.0)))))) / 2.0;
	elseif (x <= 5.2e+157)
		tmp = (eps_m * ((t_0 / eps_m) - x)) / 2.0;
	elseif (x <= 5.7e+255)
		tmp = 0.0;
	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[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.15e-232], N[(N[(2.0 + N[(x * N[(-1.0 + N[(0.5 * N[(x * N[Power[N[(eps$95$m + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.2e+157], N[(N[(eps$95$m * N[(N[(t$95$0 / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.7e+255], 0.0, N[(t$95$0 / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.1499999999999998e-232

   1. Initial program 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around 0 44.7%

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

   6. Taylor expanded in eps around inf 82.4%

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

   7. Taylor expanded in eps around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-rgt-neg-in 82.3%

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

      3. mul-1-neg 82.3%

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

      4. sub-neg 82.3%

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

      5. distribute-lft-in 82.3%

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

      6. metadata-eval 82.3%

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

      7. mul-1-neg 82.3%

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

      8. remove-double-neg 82.3%

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

   9. Simplified 82.3%

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

   10. Taylor expanded in x around 0 88.0%

       \[\leadsto \frac{\color{blue}{2 + x \cdot \left(0.5 \cdot \left(x \cdot {\left(\varepsilon - 1\right)}^{2}\right) - 1\right)}}{2} \]

   if -2.1499999999999998e-232 < x < 5.20000000000000022e157

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in eps around inf 69.1%

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

   7. Taylor expanded in eps around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. distribute-rgt-neg-in 69.1%

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

      3. mul-1-neg 69.1%

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

      4. sub-neg 69.1%

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

      5. distribute-lft-in 69.1%

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

      6. metadata-eval 69.1%

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

      7. mul-1-neg 69.1%

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

      8. remove-double-neg 69.1%

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

   9. Simplified 69.1%

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

   if 5.20000000000000022e157 < x < 5.69999999999999987e255

   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%

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

   5. Taylor expanded in eps around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. mul-1-neg 65.8%

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

      3. rec-exp 65.8%

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

      4. sub-neg 65.8%

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

      5. div-sub 65.8%

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

      6. mul-1-neg 65.8%

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

      7. rec-exp 65.8%

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

      8. +-inverses 65.8%

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

   7. Simplified 65.8%

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

   if 5.69999999999999987e255 < 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 47.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-232}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + 0.5 \cdot \left(x \cdot {\left(\varepsilon + -1\right)}^{2}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} - x\right)}{2}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+255}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-247}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+155} \lor \neg \left(x \leq 1.1 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 4e-247)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 7.2e+155) (not (<= x 1.1e+258)))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4e-247) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 7.2e+155) || !(x <= 1.1e+258)) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 4d-247) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 7.2d+155) .or. (.not. (x <= 1.1d+258))) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 4e-247) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 7.2e+155) || !(x <= 1.1e+258)) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4e-247:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 7.2e+155) or not (x <= 1.1e+258):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4e-247)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 7.2e+155) || !(x <= 1.1e+258))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 4e-247)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 7.2e+155) || ~((x <= 1.1e+258)))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 4e-247], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 7.2e+155], N[Not[LessEqual[x, 1.1e+258]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 4.0000000000000001e-247

   1. Initial program 70.1%

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

   3. Simplified 70.1%

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

   5. Taylor expanded in x around 0 49.8%

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

   6. Taylor expanded in eps around inf 86.9%

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

   7. Taylor expanded in eps around 0 85.9%

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

      1. neg-mul-1 85.9%

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

   9. Simplified 85.9%

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

   if 4.0000000000000001e-247 < x < 7.20000000000000015e155 or 1.09999999999999991e258 < x

   1. Initial program 71.7%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in eps around inf 97.2%

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

   6. Taylor expanded in x around 0 55.1%

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

   if 7.20000000000000015e155 < x < 1.09999999999999991e258

   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%

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

   5. Taylor expanded in eps around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. mul-1-neg 65.8%

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

      3. rec-exp 65.8%

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

      4. sub-neg 65.8%

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

      5. div-sub 65.8%

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

      6. mul-1-neg 65.8%

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

      7. rec-exp 65.8%

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

      8. +-inverses 65.8%

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

   7. Simplified 65.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-247}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+155} \lor \neg \left(x \leq 1.1 \cdot 10^{+258}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{eps\_m} + 0.5 \cdot \frac{-1}{eps\_m}\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 122:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{2}{eps\_m} - x\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\frac{0.5}{\frac{eps\_m}{\mathsf{expm1}\left(x\right)}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+255}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           (/
            (*
             x
             (+ 0.25 (* x (- (* x 0.020833333333333332) 0.08333333333333333))))
            eps_m)
           (* 0.5 (/ -1.0 eps_m))))))
   (if (<= x -2.4e+101)
     t_0
     (if (<= x 122.0)
       (/ (* eps_m (- (/ 2.0 eps_m) x)) 2.0)
       (if (<= x 2e+157)
         (/ 0.5 (/ eps_m (expm1 x)))
         (if (<= x 2e+255) 0.0 t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	double tmp;
	if (x <= -2.4e+101) {
		tmp = t_0;
	} else if (x <= 122.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else if (x <= 2e+157) {
		tmp = 0.5 / (eps_m / expm1(x));
	} else if (x <= 2e+255) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	double tmp;
	if (x <= -2.4e+101) {
		tmp = t_0;
	} else if (x <= 122.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else if (x <= 2e+157) {
		tmp = 0.5 / (eps_m / Math.expm1(x));
	} else if (x <= 2e+255) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)))
	tmp = 0
	if x <= -2.4e+101:
		tmp = t_0
	elif x <= 122.0:
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0
	elif x <= 2e+157:
		tmp = 0.5 / (eps_m / math.expm1(x))
	elif x <= 2e+255:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(Float64(Float64(x * Float64(0.25 + Float64(x * Float64(Float64(x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + Float64(0.5 * Float64(-1.0 / eps_m))))
	tmp = 0.0
	if (x <= -2.4e+101)
		tmp = t_0;
	elseif (x <= 122.0)
		tmp = Float64(Float64(eps_m * Float64(Float64(2.0 / eps_m) - x)) / 2.0);
	elseif (x <= 2e+157)
		tmp = Float64(0.5 / Float64(eps_m / expm1(x)));
	elseif (x <= 2e+255)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(N[(x * N[(0.25 + N[(x * N[(N[(x * 0.020833333333333332), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] + N[(0.5 * N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e+101], t$95$0, If[LessEqual[x, 122.0], N[(N[(eps$95$m * N[(N[(2.0 / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+157], N[(0.5 / N[(eps$95$m / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2e+255], 0.0, t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.39999999999999988e101 or 1.99999999999999998e255 < 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 42.8%

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

   6. Taylor expanded in eps around 0 41.6%

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

      1. expm1-define 41.6%

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

      2. mul-1-neg 41.6%

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

   8. Simplified 41.6%

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

   9. Taylor expanded in x around 0 51.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.020833333333333332 \cdot \frac{x}{\varepsilon} - 0.08333333333333333 \cdot \frac{1}{\varepsilon}\right) + 0.25 \cdot \frac{1}{\varepsilon}\right) - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]

   10. Taylor expanded in eps around 0 56.6%

       \[\leadsto x \cdot \left(\color{blue}{\frac{x \cdot \left(0.25 + x \cdot \left(0.020833333333333332 \cdot x - 0.08333333333333333\right)\right)}{\varepsilon}} - 0.5 \cdot \frac{1}{\varepsilon}\right) \]

   if -2.39999999999999988e101 < x < 122

   1. Initial program 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in x around 0 38.2%

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

   6. Taylor expanded in eps around inf 80.6%

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

   7. Taylor expanded in x around 0 68.0%

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

   if 122 < x < 1.99999999999999997e157

   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 29.6%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. mul-1-neg 1.9%

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

   8. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. clear-num 1.9%

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

      2. inv-pow 1.9%

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

      3. div-inv 1.9%

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

      4. clear-num 1.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 28.4%

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

      7. sqr-neg 28.4%

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

      8. sqrt-unprod 28.4%

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

      9. add-sqr-sqrt 28.4%

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

   10. Applied egg-rr 28.4%

       \[\leadsto \color{blue}{{\left(2 \cdot \frac{\varepsilon}{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 28.4%

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

       2. associate-/r* 28.4%

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

       3. metadata-eval 28.4%

          \[\leadsto \frac{\color{blue}{0.5}}{\frac{\varepsilon}{\mathsf{expm1}\left(x\right)}} \]

   12. Simplified 28.4%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{\varepsilon}{\mathsf{expm1}\left(x\right)}}} \]

   if 1.99999999999999997e157 < x < 1.99999999999999998e255

   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%

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

   5. Taylor expanded in eps around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. mul-1-neg 65.8%

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

      3. rec-exp 65.8%

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

      4. sub-neg 65.8%

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

      5. div-sub 65.8%

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

      6. mul-1-neg 65.8%

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

      7. rec-exp 65.8%

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

      8. +-inverses 65.8%

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

   7. Simplified 65.8%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{\varepsilon} + 0.5 \cdot \frac{-1}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 122:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{2}{\varepsilon} - x\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\frac{0.5}{\frac{\varepsilon}{\mathsf{expm1}\left(x\right)}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+255}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{\varepsilon} + 0.5 \cdot \frac{-1}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 660:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{0.5}{\frac{eps\_m}{\mathsf{expm1}\left(x\right)}}\\ \mathbf{elif}\;x \leq 10^{+254}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{eps\_m} + 0.5 \cdot \frac{-1}{eps\_m}\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 660.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 2e+156)
     (/ 0.5 (/ eps_m (expm1 x)))
     (if (<= x 1e+254)
       0.0
       (*
        x
        (+
         (/
          (*
           x
           (+ 0.25 (* x (- (* x 0.020833333333333332) 0.08333333333333333))))
          eps_m)
         (* 0.5 (/ -1.0 eps_m))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 660.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 2e+156) {
		tmp = 0.5 / (eps_m / expm1(x));
	} else if (x <= 1e+254) {
		tmp = 0.0;
	} else {
		tmp = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 660.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 2e+156) {
		tmp = 0.5 / (eps_m / Math.expm1(x));
	} else if (x <= 1e+254) {
		tmp = 0.0;
	} else {
		tmp = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 660.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 2e+156:
		tmp = 0.5 / (eps_m / math.expm1(x))
	elif x <= 1e+254:
		tmp = 0.0
	else:
		tmp = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 660.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 2e+156)
		tmp = Float64(0.5 / Float64(eps_m / expm1(x)));
	elseif (x <= 1e+254)
		tmp = 0.0;
	else
		tmp = Float64(x * Float64(Float64(Float64(x * Float64(0.25 + Float64(x * Float64(Float64(x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + Float64(0.5 * Float64(-1.0 / eps_m))));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 660.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+156], N[(0.5 / N[(eps$95$m / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e+254], 0.0, N[(x * N[(N[(N[(x * N[(0.25 + N[(x * N[(N[(x * 0.020833333333333332), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] + N[(0.5 * N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 660

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in eps around inf 81.9%

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

   7. Taylor expanded in eps around 0 78.5%

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

      1. neg-mul-1 78.5%

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

   9. Simplified 78.5%

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

   if 660 < x < 2e156

   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 29.6%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. mul-1-neg 1.9%

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

   8. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. clear-num 1.9%

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

      2. inv-pow 1.9%

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

      3. div-inv 1.9%

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

      4. clear-num 1.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 28.4%

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

      7. sqr-neg 28.4%

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

      8. sqrt-unprod 28.4%

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

      9. add-sqr-sqrt 28.4%

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

   10. Applied egg-rr 28.4%

       \[\leadsto \color{blue}{{\left(2 \cdot \frac{\varepsilon}{\mathsf{expm1}\left(x\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 28.4%

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

       2. associate-/r* 28.4%

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

       3. metadata-eval 28.4%

          \[\leadsto \frac{\color{blue}{0.5}}{\frac{\varepsilon}{\mathsf{expm1}\left(x\right)}} \]

   12. Simplified 28.4%

       \[\leadsto \color{blue}{\frac{0.5}{\frac{\varepsilon}{\mathsf{expm1}\left(x\right)}}} \]

   if 2e156 < x < 9.9999999999999994e253

   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%

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

   5. Taylor expanded in eps around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. mul-1-neg 65.8%

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

      3. rec-exp 65.8%

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

      4. sub-neg 65.8%

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

      5. div-sub 65.8%

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

      6. mul-1-neg 65.8%

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

      7. rec-exp 65.8%

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

      8. +-inverses 65.8%

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

   7. Simplified 65.8%

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

   if 9.9999999999999994e253 < 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 47.5%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

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

      2. mul-1-neg 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in x around 0 46.6%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.020833333333333332 \cdot \frac{x}{\varepsilon} - 0.08333333333333333 \cdot \frac{1}{\varepsilon}\right) + 0.25 \cdot \frac{1}{\varepsilon}\right) - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]

   10. Taylor expanded in eps around 0 46.6%

       \[\leadsto x \cdot \left(\color{blue}{\frac{x \cdot \left(0.25 + x \cdot \left(0.020833333333333332 \cdot x - 0.08333333333333333\right)\right)}{\varepsilon}} - 0.5 \cdot \frac{1}{\varepsilon}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 660:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+156}:\\ \;\;\;\;\frac{0.5}{\frac{\varepsilon}{\mathsf{expm1}\left(x\right)}}\\ \mathbf{elif}\;x \leq 10^{+254}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{\varepsilon} + 0.5 \cdot \frac{-1}{\varepsilon}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{eps\_m} + 0.5 \cdot \frac{-1}{eps\_m}\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{+97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 210:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{2}{eps\_m} - x\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+105}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+156} \lor \neg \left(x \leq 3.5 \cdot 10^{+254}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           (/
            (*
             x
             (+ 0.25 (* x (- (* x 0.020833333333333332) 0.08333333333333333))))
            eps_m)
           (* 0.5 (/ -1.0 eps_m))))))
   (if (<= x -2.7e+97)
     t_0
     (if (<= x 210.0)
       (/ (* eps_m (- (/ 2.0 eps_m) x)) 2.0)
       (if (<= x 1e+105)
         0.0
         (if (or (<= x 1.05e+156) (not (<= x 3.5e+254))) t_0 0.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	double tmp;
	if (x <= -2.7e+97) {
		tmp = t_0;
	} else if (x <= 210.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else if (x <= 1e+105) {
		tmp = 0.0;
	} else if ((x <= 1.05e+156) || !(x <= 3.5e+254)) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (((x * (0.25d0 + (x * ((x * 0.020833333333333332d0) - 0.08333333333333333d0)))) / eps_m) + (0.5d0 * ((-1.0d0) / eps_m)))
    if (x <= (-2.7d+97)) then
        tmp = t_0
    else if (x <= 210.0d0) then
        tmp = (eps_m * ((2.0d0 / eps_m) - x)) / 2.0d0
    else if (x <= 1d+105) then
        tmp = 0.0d0
    else if ((x <= 1.05d+156) .or. (.not. (x <= 3.5d+254))) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	double tmp;
	if (x <= -2.7e+97) {
		tmp = t_0;
	} else if (x <= 210.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else if (x <= 1e+105) {
		tmp = 0.0;
	} else if ((x <= 1.05e+156) || !(x <= 3.5e+254)) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)))
	tmp = 0
	if x <= -2.7e+97:
		tmp = t_0
	elif x <= 210.0:
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0
	elif x <= 1e+105:
		tmp = 0.0
	elif (x <= 1.05e+156) or not (x <= 3.5e+254):
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(Float64(Float64(x * Float64(0.25 + Float64(x * Float64(Float64(x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + Float64(0.5 * Float64(-1.0 / eps_m))))
	tmp = 0.0
	if (x <= -2.7e+97)
		tmp = t_0;
	elseif (x <= 210.0)
		tmp = Float64(Float64(eps_m * Float64(Float64(2.0 / eps_m) - x)) / 2.0);
	elseif (x <= 1e+105)
		tmp = 0.0;
	elseif ((x <= 1.05e+156) || !(x <= 3.5e+254))
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (((x * (0.25 + (x * ((x * 0.020833333333333332) - 0.08333333333333333)))) / eps_m) + (0.5 * (-1.0 / eps_m)));
	tmp = 0.0;
	if (x <= -2.7e+97)
		tmp = t_0;
	elseif (x <= 210.0)
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	elseif (x <= 1e+105)
		tmp = 0.0;
	elseif ((x <= 1.05e+156) || ~((x <= 3.5e+254)))
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(N[(N[(x * N[(0.25 + N[(x * N[(N[(x * 0.020833333333333332), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] + N[(0.5 * N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e+97], t$95$0, If[LessEqual[x, 210.0], N[(N[(eps$95$m * N[(N[(2.0 / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+105], 0.0, If[Or[LessEqual[x, 1.05e+156], N[Not[LessEqual[x, 3.5e+254]], $MachinePrecision]], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.69999999999999993e97 or 9.9999999999999994e104 < x < 1.04999999999999991e156 or 3.50000000000000017e254 < 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 42.4%

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

   6. Taylor expanded in eps around 0 30.6%

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

      1. expm1-define 30.6%

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

      2. mul-1-neg 30.6%

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

   8. Simplified 30.6%

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

   9. Taylor expanded in x around 0 46.9%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.020833333333333332 \cdot \frac{x}{\varepsilon} - 0.08333333333333333 \cdot \frac{1}{\varepsilon}\right) + 0.25 \cdot \frac{1}{\varepsilon}\right) - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]

   10. Taylor expanded in eps around 0 52.1%

       \[\leadsto x \cdot \left(\color{blue}{\frac{x \cdot \left(0.25 + x \cdot \left(0.020833333333333332 \cdot x - 0.08333333333333333\right)\right)}{\varepsilon}} - 0.5 \cdot \frac{1}{\varepsilon}\right) \]

   if -2.69999999999999993e97 < x < 210

   1. Initial program 57.0%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in x around 0 38.2%

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

   6. Taylor expanded in eps around inf 80.6%

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

   7. Taylor expanded in x around 0 68.0%

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

   if 210 < x < 9.9999999999999994e104 or 1.04999999999999991e156 < x < 3.50000000000000017e254

   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%

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

   5. Taylor expanded in eps around 0 57.5%

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

      1. mul-1-neg 57.5%

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

      2. mul-1-neg 57.5%

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

      3. rec-exp 57.5%

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

      4. sub-neg 57.5%

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

      5. div-sub 57.5%

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

      6. mul-1-neg 57.5%

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

      7. rec-exp 57.5%

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

      8. +-inverses 57.5%

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

   7. Simplified 57.5%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+97}:\\ \;\;\;\;x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{\varepsilon} + 0.5 \cdot \frac{-1}{\varepsilon}\right)\\ \mathbf{elif}\;x \leq 210:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{2}{\varepsilon} - x\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+105}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+156} \lor \neg \left(x \leq 3.5 \cdot 10^{+254}\right):\\ \;\;\;\;x \cdot \left(\frac{x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332 - 0.08333333333333333\right)\right)}{\varepsilon} + 0.5 \cdot \frac{-1}{\varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.6% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 260:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{2}{eps\_m} - x\right)}{2}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+257}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - eps\_m\right) \cdot \left(-1 + \frac{-1}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 260.0)
   (/ (* eps_m (- (/ 2.0 eps_m) x)) 2.0)
   (if (<= x 2.4e+257)
     0.0
     (/ (+ 2.0 (* x (* (- 1.0 eps_m) (+ -1.0 (/ -1.0 eps_m))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 260.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else if (x <= 2.4e+257) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 260.0d0) then
        tmp = (eps_m * ((2.0d0 / eps_m) - x)) / 2.0d0
    else if (x <= 2.4d+257) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 - eps_m) * ((-1.0d0) + ((-1.0d0) / eps_m))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 260.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else if (x <= 2.4e+257) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 260.0:
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0
	elif x <= 2.4e+257:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 260.0)
		tmp = Float64(Float64(eps_m * Float64(Float64(2.0 / eps_m) - x)) / 2.0);
	elseif (x <= 2.4e+257)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - eps_m) * Float64(-1.0 + Float64(-1.0 / eps_m))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 260.0)
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	elseif (x <= 2.4e+257)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * ((1.0 - eps_m) * (-1.0 + (-1.0 / eps_m))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 260.0], N[(N[(eps$95$m * N[(N[(2.0 / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.4e+257], 0.0, N[(N[(2.0 + N[(x * N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 260

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in eps around inf 81.9%

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

   7. Taylor expanded in x around 0 65.6%

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

   if 260 < x < 2.4000000000000001e257

   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%

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

   5. Taylor expanded in eps around 0 51.6%

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

      1. mul-1-neg 51.6%

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

      2. mul-1-neg 51.6%

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

      3. rec-exp 51.6%

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

      4. sub-neg 51.6%

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

      5. div-sub 51.6%

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

      6. mul-1-neg 51.6%

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

      7. rec-exp 51.6%

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

      8. +-inverses 51.6%

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

   7. Simplified 51.6%

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

   if 2.4000000000000001e257 < 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 47.5%

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

   6. Taylor expanded in x around 0 46.6%

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

      1. mul-1-neg 46.6%

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

      2. distribute-rgt-neg-in 46.6%

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

      3. *-commutative 46.6%

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

      4. distribute-rgt-neg-in 46.6%

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

      5. mul-1-neg 46.6%

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

      6. distribute-lft-in 46.6%

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

      7. metadata-eval 46.6%

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

      8. neg-mul-1 46.6%

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

      9. distribute-neg-frac 46.6%

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

      10. metadata-eval 46.6%

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

   8. Simplified 46.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 260:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{2}{\varepsilon} - x\right)}{2}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+257}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(1 - \varepsilon\right) \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.4% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 255:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{2}{eps\_m} - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 255.0) (/ (* eps_m (- (/ 2.0 eps_m) x)) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 255.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 255.0d0) then
        tmp = (eps_m * ((2.0d0 / eps_m) - x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 255.0) {
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 255.0:
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 255.0)
		tmp = Float64(Float64(eps_m * Float64(Float64(2.0 / eps_m) - x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 255.0)
		tmp = (eps_m * ((2.0 / eps_m) - x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 255.0], N[(N[(eps$95$m * N[(N[(2.0 / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 255

   1. Initial program 63.1%

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

   3. Simplified 63.1%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in eps around inf 81.9%

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

   7. Taylor expanded in x around 0 65.6%

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

   if 255 < 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%

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

   5. Taylor expanded in eps around 0 48.0%

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

      1. mul-1-neg 48.0%

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

      2. mul-1-neg 48.0%

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

      3. rec-exp 48.0%

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

      4. sub-neg 48.0%

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

      5. div-sub 48.0%

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

      6. mul-1-neg 48.0%

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

      7. rec-exp 48.0%

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

      8. +-inverses 48.0%

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

   7. Simplified 48.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 255:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{2}{\varepsilon} - x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.7% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot eps\_m}{-2}\\ \mathbf{elif}\;x \leq 1200000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (/ (* x eps_m) (- 2.0)) (if (<= x 1200000.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) / -2.0;
	} else if (x <= 1200000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) / -2.0d0
    else if (x <= 1200000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) / -2.0;
	} else if (x <= 1200000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) / -2.0
	elif x <= 1200000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) / Float64(-2.0));
	elseif (x <= 1200000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) / -2.0;
	elseif (x <= 1200000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] / (-2.0)), $MachinePrecision], If[LessEqual[x, 1200000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   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 57.7%

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

   6. Taylor expanded in eps around inf 37.3%

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

      1. associate-*r* 37.3%

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

      2. mul-1-neg 37.3%

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

   8. Simplified 37.3%

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

   if -1 < x < 1.2e6

   1. Initial program 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in x around 0 73.4%

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

   if 1.2e6 < 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%

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

   5. Taylor expanded in eps around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. mul-1-neg 48.7%

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

      3. rec-exp 48.7%

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

      4. sub-neg 48.7%

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

      5. div-sub 48.7%

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

      6. mul-1-neg 48.7%

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

      7. rec-exp 48.7%

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

      8. +-inverses 48.7%

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

   7. Simplified 48.7%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \varepsilon}{-2}\\ \mathbf{elif}\;x \leq 1200000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.1% accurate, 22.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in x around 0 41.8%

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

   6. Taylor expanded in eps around inf 82.2%

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

   7. Taylor expanded in eps around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. distribute-rgt-neg-in 82.2%

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

      3. mul-1-neg 82.2%

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

      4. sub-neg 82.2%

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

      5. distribute-lft-in 82.2%

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

      6. metadata-eval 82.2%

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

      7. mul-1-neg 82.2%

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

      8. remove-double-neg 82.2%

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

   9. Simplified 82.2%

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

   10. Taylor expanded in x around 0 59.2%

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

       1. neg-mul-1 59.2%

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

       2. unsub-neg 59.2%

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

   12. Simplified 59.2%

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

   if 2 < x

   1. Initial program 98.7%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in eps around 0 47.4%

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

      1. mul-1-neg 47.4%

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

      2. mul-1-neg 47.4%

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

      3. rec-exp 47.4%

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

      4. sub-neg 47.4%

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

      5. div-sub 47.4%

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

      6. mul-1-neg 47.4%

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

      7. rec-exp 47.4%

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

      8. +-inverses 47.4%

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

   7. Simplified 47.4%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.8% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1200000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 1200000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1200000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1200000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1200000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1200000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1200000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1200000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1200000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e6

   1. Initial program 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in x around 0 58.6%

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

   if 1.2e6 < 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%

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

   5. Taylor expanded in eps around 0 48.7%

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

      1. mul-1-neg 48.7%

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

      2. mul-1-neg 48.7%

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

      3. rec-exp 48.7%

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

      4. sub-neg 48.7%

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

      5. div-sub 48.7%

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

      6. mul-1-neg 48.7%

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

      7. rec-exp 48.7%

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

      8. +-inverses 48.7%

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

   7. Simplified 48.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1200000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 16.7% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

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

3. Simplified 61.0%

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

5. Taylor expanded in eps around 0 15.1%

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

   1. mul-1-neg 15.1%

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

   2. mul-1-neg 15.1%

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

   3. rec-exp 15.1%

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

   4. sub-neg 15.1%

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

   5. div-sub 15.1%

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

   6. mul-1-neg 15.1%

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

   7. rec-exp 15.1%

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

   8. +-inverses 15.3%

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

7. Simplified 15.3%

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

8. Final simplification 15.3%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024074 
(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))