NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.1% → 99.8%

Time: 13.8s

Alternatives: 12

Speedup: 2.0×

• 38.4% 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: 74.1% 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: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.45e-5)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x eps_m)) (exp (* eps_m (- x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.45e-5) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5.45d-5) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((eps_m * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.45e-5) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5.45e-5:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5.45e-5)
		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 * eps_m)) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5.45e-5)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * eps_m)) + exp((eps_m * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5.45e-5], 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 * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.45000000000000034e-5

   1. Initial program 61.6%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in eps around 0 31.0%

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

      1. associate-+r+ 70.7%

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

      2. mul-1-neg 70.7%

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

      3. sub-neg 70.7%

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

      4. +-inverses 70.7%

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

      5. associate-*r* 70.7%

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

      6. distribute-rgt-out 71.3%

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

      7. mul-1-neg 71.3%

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

   7. Simplified 71.3%

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

   if 5.45000000000000034e-5 < 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 87.6%

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

   5. Taylor expanded in eps around inf 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. cancel-sign-sub-inv 100.0%

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

      2. exp-sum 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-lft-identity 100.0%

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

      5. exp-sum 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. +-commutative 100.0%

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

      8. remove-double-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. sub-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. exp-neg 100.0%

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

      13. distribute-rgt-neg-in 100.0%

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

      14. cancel-sign-sub-inv 100.0%

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

      15. metadata-eval 100.0%

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

      16. *-lft-identity 100.0%

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

      17. distribute-neg-in 100.0%

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

      18. metadata-eval 100.0%

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

      19. 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%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. mul-1-neg 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

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 71.2%

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

3. Simplified 64.4%

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

5. Taylor expanded in eps around inf 98.5%

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

6. Taylor expanded in x around -inf 98.5%

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

   1. cancel-sign-sub-inv 98.5%

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

   2. exp-sum 79.7%

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

   3. metadata-eval 79.7%

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

   4. *-lft-identity 79.7%

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

   5. exp-sum 98.5%

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

   6. distribute-lft1-in 98.5%

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

   7. +-commutative 98.5%

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

   8. remove-double-neg 98.5%

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

   9. mul-1-neg 98.5%

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

   10. sub-neg 98.5%

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

   11. *-commutative 98.5%

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

   12. exp-neg 98.5%

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

   13. distribute-rgt-neg-in 98.5%

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

   14. cancel-sign-sub-inv 98.5%

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

   15. metadata-eval 98.5%

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

   16. *-lft-identity 98.5%

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

   17. distribute-neg-in 98.5%

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

   18. metadata-eval 98.5%

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

   19. unsub-neg 98.5%

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 3.3 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{1 + e^{eps\_m \cdot \left(-x\right)}}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.45e-5)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (if (<= eps_m 3.3e+126)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (/ (* eps_m (+ x (/ (+ 1.0 (exp (* eps_m (- x)))) eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.45e-5) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 3.3e+126) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = (eps_m * (x + ((1.0 + exp((eps_m * -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 <= 5.45d-5) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else if (eps_m <= 3.3d+126) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = (eps_m * (x + ((1.0d0 + exp((eps_m * -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 <= 5.45e-5) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else if (eps_m <= 3.3e+126) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = (eps_m * (x + ((1.0 + Math.exp((eps_m * -x))) / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5.45e-5:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	elif eps_m <= 3.3e+126:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = (eps_m * (x + ((1.0 + math.exp((eps_m * -x))) / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5.45e-5)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	elseif (eps_m <= 3.3e+126)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5.45e-5)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	elseif (eps_m <= 3.3e+126)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = (eps_m * (x + ((1.0 + exp((eps_m * -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, 5.45e-5], 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, 3.3e+126], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * N[(x + N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 5.45000000000000034e-5

   1. Initial program 61.6%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in eps around 0 31.0%

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

      1. associate-+r+ 70.7%

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

      2. mul-1-neg 70.7%

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

      3. sub-neg 70.7%

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

      4. +-inverses 70.7%

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

      5. associate-*r* 70.7%

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

      6. distribute-rgt-out 71.3%

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

      7. mul-1-neg 71.3%

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

   7. Simplified 71.3%

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

   if 5.45000000000000034e-5 < eps < 3.30000000000000013e126

   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 95.6%

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

   5. Taylor expanded in eps around inf 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 86.8%

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

   7. Taylor expanded in eps around inf 86.8%

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

      1. *-commutative 100.0%

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

   9. Simplified 86.8%

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

   if 3.30000000000000013e126 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. associate-+r+ 65.9%

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

      3. mul-1-neg 65.9%

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

      4. distribute-rgt-neg-in 65.9%

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

      5. *-commutative 65.9%

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

      6. distribute-rgt-neg-in 65.9%

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

      7. neg-mul-1 65.9%

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

      8. distribute-rgt-in 65.9%

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

      9. metadata-eval 65.9%

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

      10. associate-*l/ 65.9%

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

      11. metadata-eval 65.9%

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

   7. Simplified 65.9%

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

   8. Taylor expanded in eps around inf 80.2%

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

   10. Simplified 80.2%

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

   11. Taylor expanded in eps around inf 80.2%

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

       1. associate-*r* 100.0%

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

       2. mul-1-neg 100.0%

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

   13. Simplified 80.2%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.45 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.3 \cdot 10^{+126}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-305}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{1 + e^{eps\_m \cdot \left(-x\right)}}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 8e-305)
   (/ (* eps_m (+ x (/ (+ 1.0 (exp (* eps_m (- x)))) eps_m))) 2.0)
   (/ (+ 1.0 (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 8e-305) {
		tmp = (eps_m * (x + ((1.0 + exp((eps_m * -x))) / eps_m))) / 2.0;
	} else {
		tmp = (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 (x <= 8d-305) then
        tmp = (eps_m * (x + ((1.0d0 + exp((eps_m * -x))) / eps_m))) / 2.0d0
    else
        tmp = (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 (x <= 8e-305) {
		tmp = (eps_m * (x + ((1.0 + Math.exp((eps_m * -x))) / eps_m))) / 2.0;
	} else {
		tmp = (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 x <= 8e-305:
		tmp = (eps_m * (x + ((1.0 + math.exp((eps_m * -x))) / eps_m))) / 2.0
	else:
		tmp = (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 (x <= 8e-305)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 8e-305)
		tmp = (eps_m * (x + ((1.0 + exp((eps_m * -x))) / eps_m))) / 2.0;
	else
		tmp = (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[x, 8e-305], N[(N[(eps$95$m * N[(x + N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999997e-305

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around 0 40.6%

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

      1. +-commutative 40.6%

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

      2. associate-+r+ 40.6%

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

      3. mul-1-neg 40.6%

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

      4. distribute-rgt-neg-in 40.6%

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

      5. *-commutative 40.6%

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

      6. distribute-rgt-neg-in 40.6%

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

      7. neg-mul-1 40.6%

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

      8. distribute-rgt-in 40.6%

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

      9. metadata-eval 40.6%

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

      10. associate-*l/ 40.6%

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

      11. metadata-eval 40.6%

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

   7. Simplified 40.6%

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

   8. Taylor expanded in eps around inf 78.3%

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

   10. Simplified 78.3%

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

   11. Taylor expanded in eps around inf 78.4%

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

       1. associate-*r* 98.1%

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

       2. mul-1-neg 98.1%

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

   13. Simplified 78.4%

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

   if 7.99999999999999997e-305 < x

   1. Initial program 75.0%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 63.0%

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

   7. Taylor expanded in eps around inf 63.3%

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

      1. *-commutative 84.7%

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

   9. Simplified 63.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-305}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-298}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-298)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ 1.0 (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-298) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (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 (x <= (-2d-298)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (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 (x <= -2e-298) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (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 x <= -2e-298:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (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 (x <= -2e-298)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-298)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (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[x, -2e-298], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999982e-298

   1. Initial program 66.1%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in eps around inf 97.9%

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

   6. Taylor expanded in x around -inf 97.9%

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

      1. cancel-sign-sub-inv 97.9%

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

      2. exp-sum 78.5%

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

      3. metadata-eval 78.5%

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

      4. *-lft-identity 78.5%

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

      5. exp-sum 97.9%

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

      6. distribute-lft1-in 97.9%

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

      7. +-commutative 97.9%

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

      8. remove-double-neg 97.9%

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

      9. mul-1-neg 97.9%

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

      10. sub-neg 97.9%

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

      11. *-commutative 97.9%

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

      12. exp-neg 97.9%

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

      13. distribute-rgt-neg-in 97.9%

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

      14. cancel-sign-sub-inv 97.9%

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

      15. metadata-eval 97.9%

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

      16. *-lft-identity 97.9%

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

      17. distribute-neg-in 97.9%

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

      18. metadata-eval 97.9%

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

      19. unsub-neg 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in eps around inf 97.9%

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

       1. *-commutative 97.9%

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

   11. Simplified 97.9%

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

   12. Taylor expanded in x around 0 68.1%

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

   if -1.99999999999999982e-298 < x

   1. Initial program 74.7%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 63.2%

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

   7. Taylor expanded in eps around inf 63.5%

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

      1. *-commutative 84.9%

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

   9. Simplified 63.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-298}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -70000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -70000000.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (/ (+ 1.0 (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -70000000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -70000000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else {
		tmp = (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 x <= -70000000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	else:
		tmp = (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 (x <= -70000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -70000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7e7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 60.1%

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

   6. Taylor expanded in eps around 0 41.2%

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

      1. expm1-define 41.2%

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

      2. neg-mul-1 41.2%

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

   8. Simplified 41.2%

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

   if -7e7 < x

   1. Initial program 66.8%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in x around 0 72.0%

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

   7. Taylor expanded in eps around inf 72.3%

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

      1. *-commutative 88.6%

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

   9. Simplified 72.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -70000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -70000000.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 4000.0) 1.0 (if (<= x 8.2e+229) 0.0 (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -70000000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 4000.0) {
		tmp = 1.0;
	} else if (x <= 8.2e+229) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -70000000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 4000.0) {
		tmp = 1.0;
	} else if (x <= 8.2e+229) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -70000000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 4000.0:
		tmp = 1.0
	elif x <= 8.2e+229:
		tmp = 0.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -70000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 4000.0)
		tmp = 1.0;
	elseif (x <= 8.2e+229)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -70000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4000.0], 1.0, If[LessEqual[x, 8.2e+229], 0.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7e7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 60.1%

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

   6. Taylor expanded in eps around 0 41.2%

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

      1. expm1-define 41.2%

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

      2. neg-mul-1 41.2%

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

   8. Simplified 41.2%

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

   if -7e7 < x < 4e3

   1. Initial program 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in x around 0 78.7%

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

   if 4e3 < x < 8.2000000000000003e229

   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 50.8%

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

      1. mul-1-neg 50.8%

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

      2. mul-1-neg 50.8%

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

      3. rec-exp 50.8%

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

      4. sub-neg 50.8%

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

      5. div-sub 50.8%

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

      6. mul-1-neg 50.8%

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

      7. rec-exp 50.8%

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

      8. +-inverses 50.8%

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

   7. Simplified 50.8%

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

   if 8.2000000000000003e229 < 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 53.9%

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

      1. +-commutative 53.9%

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

      2. associate-+r+ 53.9%

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

      3. mul-1-neg 53.9%

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

      4. distribute-rgt-neg-in 53.9%

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

      5. *-commutative 53.9%

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

      6. distribute-rgt-neg-in 53.9%

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

      7. neg-mul-1 53.9%

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

      8. distribute-rgt-in 53.9%

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

      9. metadata-eval 53.9%

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

      10. associate-*l/ 53.9%

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

      11. metadata-eval 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in eps around inf 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in eps around inf 48.9%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 48.9%

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

   13. Simplified 48.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* x eps_m) -0.5)
   (if (<= x 4000.0) 1.0 (if (<= x 8e+229) 0.0 (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 4000.0) {
		tmp = 1.0;
	} else if (x <= 8e+229) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 4000.0d0) then
        tmp = 1.0d0
    else if (x <= 8d+229) then
        tmp = 0.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 4000.0) {
		tmp = 1.0;
	} else if (x <= 8e+229) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) * -0.5
	elif x <= 4000.0:
		tmp = 1.0
	elif x <= 8e+229:
		tmp = 0.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 4000.0)
		tmp = 1.0;
	elseif (x <= 8e+229)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 4000.0)
		tmp = 1.0;
	elseif (x <= 8e+229)
		tmp = 0.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 4000.0], 1.0, If[LessEqual[x, 8e+229], 0.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in x around 0 50.6%

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

      1. +-commutative 50.6%

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

      2. associate-+r+ 50.6%

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

      3. mul-1-neg 50.6%

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

      4. distribute-rgt-neg-in 50.6%

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

      5. *-commutative 50.6%

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

      6. distribute-rgt-neg-in 50.6%

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

      7. neg-mul-1 50.6%

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

      8. distribute-rgt-in 50.6%

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

      9. metadata-eval 50.6%

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

      10. associate-*l/ 50.6%

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

      11. metadata-eval 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in eps around inf 66.9%

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

   10. Simplified 66.9%

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

   11. Taylor expanded in eps around inf 29.1%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 29.1%

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

   13. Simplified 29.1%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   14. Step-by-step derivation

       1. frac-2neg 29.1%

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

       2. *-commutative 29.1%

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

       3. distribute-lft-neg-out 29.1%

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

       4. div-inv 29.1%

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

       5. *-commutative 29.1%

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

       6. add-sqr-sqrt 29.0%

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

       7. sqrt-unprod 76.9%

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

       8. sqr-neg 76.9%

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

       9. sqrt-unprod 20.3%

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

       10. add-sqr-sqrt 20.5%

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

       11. metadata-eval 20.5%

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

       12. metadata-eval 20.5%

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

   15. Applied egg-rr 20.5%

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

   if -1 < x < 4e3

   1. Initial program 50.9%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in x around 0 80.2%

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

   if 4e3 < x < 7.9999999999999999e229

   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 50.8%

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

      1. mul-1-neg 50.8%

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

      2. mul-1-neg 50.8%

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

      3. rec-exp 50.8%

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

      4. sub-neg 50.8%

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

      5. div-sub 50.8%

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

      6. mul-1-neg 50.8%

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

      7. rec-exp 50.8%

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

      8. +-inverses 50.8%

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

   7. Simplified 50.8%

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

   if 7.9999999999999999e229 < 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 53.9%

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

      1. +-commutative 53.9%

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

      2. associate-+r+ 53.9%

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

      3. mul-1-neg 53.9%

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

      4. distribute-rgt-neg-in 53.9%

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

      5. *-commutative 53.9%

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

      6. distribute-rgt-neg-in 53.9%

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

      7. neg-mul-1 53.9%

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

      8. distribute-rgt-in 53.9%

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

      9. metadata-eval 53.9%

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

      10. associate-*l/ 53.9%

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

      11. metadata-eval 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in eps around inf 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in eps around inf 48.9%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 48.9%

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

   13. Simplified 48.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 300:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 300.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (<= x 8.2e+229) 0.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 300.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 8.2e+229) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 300.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= 8.2d+229) then
        tmp = 0.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 300.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 8.2e+229) {
		tmp = 0.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 300.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= 8.2e+229:
		tmp = 0.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 300.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= 8.2e+229)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 300.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= 8.2e+229)
		tmp = 0.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 300.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.2e+229], 0.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 300

   1. Initial program 59.5%

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

   3. Simplified 49.9%

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

   5. Taylor expanded in x around 0 65.0%

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

   6. Taylor expanded in eps around 0 68.1%

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

   7. Taylor expanded in x around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. *-commutative 68.1%

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

      3. unsub-neg 68.1%

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

   9. Simplified 68.1%

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

   if 300 < x < 8.2000000000000003e229

   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 49.9%

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

      1. mul-1-neg 49.9%

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

      2. mul-1-neg 49.9%

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

      3. rec-exp 49.9%

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

      4. sub-neg 49.9%

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

      5. div-sub 49.9%

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

      6. mul-1-neg 49.9%

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

      7. rec-exp 49.9%

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

      8. +-inverses 49.9%

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

   7. Simplified 49.9%

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

   if 8.2000000000000003e229 < 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 53.9%

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

      1. +-commutative 53.9%

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

      2. associate-+r+ 53.9%

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

      3. mul-1-neg 53.9%

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

      4. distribute-rgt-neg-in 53.9%

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

      5. *-commutative 53.9%

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

      6. distribute-rgt-neg-in 53.9%

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

      7. neg-mul-1 53.9%

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

      8. distribute-rgt-in 53.9%

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

      9. metadata-eval 53.9%

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

      10. associate-*l/ 53.9%

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

      11. metadata-eval 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in eps around inf 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in eps around inf 48.9%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 48.9%

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

   13. Simplified 48.9%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 300:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+229}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.9% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0) (* (* x eps_m) -0.5) (if (<= x 4000.0) 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) * -0.5;
	} else if (x <= 4000.0) {
		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) * (-0.5d0)
    else if (x <= 4000.0d0) 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) * -0.5;
	} else if (x <= 4000.0) {
		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) * -0.5
	elif x <= 4000.0:
		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) * -0.5);
	elseif (x <= 4000.0)
		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) * -0.5;
	elseif (x <= 4000.0)
		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] * -0.5), $MachinePrecision], If[LessEqual[x, 4000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in x around 0 50.6%

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

      1. +-commutative 50.6%

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

      2. associate-+r+ 50.6%

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

      3. mul-1-neg 50.6%

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

      4. distribute-rgt-neg-in 50.6%

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

      5. *-commutative 50.6%

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

      6. distribute-rgt-neg-in 50.6%

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

      7. neg-mul-1 50.6%

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

      8. distribute-rgt-in 50.6%

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

      9. metadata-eval 50.6%

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

      10. associate-*l/ 50.6%

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

      11. metadata-eval 50.6%

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

   7. Simplified 50.6%

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

   8. Taylor expanded in eps around inf 66.9%

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

   10. Simplified 66.9%

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

   11. Taylor expanded in eps around inf 29.1%

       \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   12. Step-by-step derivation

       1. *-commutative 29.1%

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

   13. Simplified 29.1%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
   14. Step-by-step derivation

       1. frac-2neg 29.1%

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

       2. *-commutative 29.1%

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

       3. distribute-lft-neg-out 29.1%

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

       4. div-inv 29.1%

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

       5. *-commutative 29.1%

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

       6. add-sqr-sqrt 29.0%

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

       7. sqrt-unprod 76.9%

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

       8. sqr-neg 76.9%

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

       9. sqrt-unprod 20.3%

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

       10. add-sqr-sqrt 20.5%

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

       11. metadata-eval 20.5%

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

       12. metadata-eval 20.5%

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

   15. Applied egg-rr 20.5%

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

   if -1 < x < 4e3

   1. Initial program 50.9%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in x around 0 80.2%

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

   if 4e3 < 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 43.4%

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

      1. mul-1-neg 43.4%

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

      2. mul-1-neg 43.4%

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

      3. rec-exp 43.4%

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

      4. sub-neg 43.4%

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

      5. div-sub 43.4%

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

      6. mul-1-neg 43.4%

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

      7. rec-exp 43.4%

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

      8. +-inverses 43.4%

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

   7. Simplified 43.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.8% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 4000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 4000.0) {
		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 <= 4000.0d0) 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 <= 4000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 4000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 4000.0)
		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 <= 4000.0)
		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, 4000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4e3

   1. Initial program 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in x around 0 64.6%

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

   if 4e3 < 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 43.4%

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

      1. mul-1-neg 43.4%

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

      2. mul-1-neg 43.4%

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

      3. rec-exp 43.4%

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

      4. sub-neg 43.4%

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

      5. div-sub 43.4%

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

      6. mul-1-neg 43.4%

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

      7. rec-exp 43.4%

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

      8. +-inverses 43.4%

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

   7. Simplified 43.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 16.9% 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 71.2%

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

3. Simplified 58.7%

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

5. Taylor expanded in eps around 0 14.0%

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

   1. mul-1-neg 14.0%

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

   2. mul-1-neg 14.0%

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

   3. rec-exp 14.0%

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

   4. sub-neg 14.0%

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

   5. div-sub 14.0%

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

   6. mul-1-neg 14.0%

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

   7. rec-exp 14.0%

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

   8. +-inverses 14.2%

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

7. Simplified 14.2%

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

8. Final simplification 14.2%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(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))