NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.8% → 99.9%

Time: 13.0s

Alternatives: 16

Speedup: 1.9×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -7.080823882121788e+285
  2. eps = 2.9295857816602393e-84

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 63.1%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in eps around 0 28.6%

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

      1. associate-+r+ 66.3%

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

      2. mul-1-neg 66.3%

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

      3. sub-neg 66.3%

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

      4. +-inverses 66.3%

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

      5. distribute-lft-out 66.3%

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

      6. distribute-rgt1-in 66.3%

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

      7. mul-1-neg 66.3%

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

   7. Simplified 66.3%

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

   if 1 < 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 92.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around -inf 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-rgt-out-- 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. *-commutative 100.0%

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

       3. distribute-rgt-neg-in 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 75.5%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

3. Simplified 67.2%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Final simplification 99.1%

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

Alternative 3: 68.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(eps\_m + 1\right) \cdot \left(1 + \left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - eps\_m\right)\right)\right)}{eps\_m + 1}}{2}\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -720.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x -5e-211)
     (/
      (+
       2.0
       (*
        x
        (/
         (* (+ eps_m 1.0) (+ 1.0 (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) eps_m))))
         (+ eps_m 1.0))))
      2.0)
     (if (<= x 5800000.0)
       1.0
       (if (<= x 9.5e+68) 0.0 (/ (/ (expm1 x) eps_m) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -720.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -5e-211) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 9.5e+68) {
		tmp = 0.0;
	} else {
		tmp = (expm1(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 <= -720.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -5e-211) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 9.5e+68) {
		tmp = 0.0;
	} else {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -720.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -5e-211:
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0
	elif x <= 5800000.0:
		tmp = 1.0
	elif x <= 9.5e+68:
		tmp = 0.0
	else:
		tmp = (math.expm1(x) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -720.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -5e-211)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps_m + 1.0) * Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - eps_m)))) / Float64(eps_m + 1.0)))) / 2.0);
	elseif (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 9.5e+68)
		tmp = 0.0;
	else
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -720.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5e-211], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5800000.0], 1.0, If[LessEqual[x, 9.5e+68], 0.0, N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -720

   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 62.7%

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

   6. Taylor expanded in eps around 0 38.5%

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

      1. expm1-define 38.5%

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

      2. neg-mul-1 38.5%

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

   8. Simplified 38.5%

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

   if -720 < x < -5.0000000000000002e-211

   1. Initial program 62.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in x around 0 57.4%

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

   6. Taylor expanded in eps around 0 56.5%

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

      1. associate--l+ 56.5%

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

      2. flip-+ 55.5%

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

      3. *-commutative 55.5%

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

      4. *-commutative 55.5%

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

      5. swap-sqr 55.5%

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

      6. metadata-eval 55.5%

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

      7. *-un-lft-identity 55.5%

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

      8. pow2 55.5%

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

   8. Applied egg-rr 36.0%

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

      1. unpow2 36.0%

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

      2. difference-of-squares 36.0%

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

      3. associate-+l+ 36.0%

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

      4. associate--l+ 36.9%

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

      5. associate--r- 36.9%

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

      6. +-inverses 36.9%

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

      7. associate-+l+ 36.9%

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

      8. metadata-eval 36.9%

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

      9. +-commutative 36.9%

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

      10. associate--l+ 52.9%

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

      11. associate--r- 52.9%

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

      12. +-inverses 52.9%

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

      13. associate-+l+ 52.9%

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

      14. metadata-eval 52.9%

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

      15. +-commutative 52.9%

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

   10. Simplified 52.9%

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

   if -5.0000000000000002e-211 < x < 5.8e6

   1. Initial program 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 80.3%

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

   if 5.8e6 < x < 9.50000000000000069e68

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 73.8%

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

      1. div-sub 73.8%

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

      2. mul-1-neg 73.8%

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

      3. rec-exp 73.8%

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

      4. +-inverses 73.8%

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

      5. metadata-eval 73.8%

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

   7. Simplified 73.8%

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

   if 9.50000000000000069e68 < 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 34.5%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 33.3%

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

      4. sqr-neg 33.3%

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

      5. sqrt-unprod 33.3%

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

      6. add-sqr-sqrt 33.3%

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

   10. Applied egg-rr 33.3%

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

       1. *-lft-identity 33.3%

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

   12. Simplified 33.3%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(\varepsilon + 1\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - \varepsilon\right)\right)\right)}{\varepsilon + 1}}{2}\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+68}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(eps\_m + 1\right) \cdot \left(1 + \left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - eps\_m\right)\right)\right)}{eps\_m + 1}}{2}\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+69}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.6e+77)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps_m)
    2.0)
   (if (<= x -5.4e-211)
     (/
      (+
       2.0
       (*
        x
        (/
         (* (+ eps_m 1.0) (+ 1.0 (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) eps_m))))
         (+ eps_m 1.0))))
      2.0)
     (if (<= x 5800000.0)
       1.0
       (if (<= x 9.2e+69) 0.0 (/ (/ (expm1 x) eps_m) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.6e+77) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -5.4e-211) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 9.2e+69) {
		tmp = 0.0;
	} else {
		tmp = (expm1(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 <= -2.6e+77) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else if (x <= -5.4e-211) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 9.2e+69) {
		tmp = 0.0;
	} else {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.6e+77:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	elif x <= -5.4e-211:
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0
	elif x <= 5800000.0:
		tmp = 1.0
	elif x <= 9.2e+69:
		tmp = 0.0
	else:
		tmp = (math.expm1(x) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0);
	elseif (x <= -5.4e-211)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps_m + 1.0) * Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - eps_m)))) / Float64(eps_m + 1.0)))) / 2.0);
	elseif (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 9.2e+69)
		tmp = 0.0;
	else
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.6e+77], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5.4e-211], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5800000.0], 1.0, If[LessEqual[x, 9.2e+69], 0.0, N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.6000000000000002e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 63.7%

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

   6. Taylor expanded in eps around 0 37.5%

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

      1. expm1-define 37.5%

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

      2. neg-mul-1 37.5%

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

   8. Simplified 37.5%

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

   9. Taylor expanded in x around 0 37.5%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -2.6000000000000002e77 < x < -5.3999999999999998e-211

   1. Initial program 71.4%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around 0 43.3%

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

      1. associate--l+ 43.3%

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

      2. flip-+ 48.6%

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

      3. *-commutative 48.6%

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

      4. *-commutative 48.6%

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

      5. swap-sqr 48.6%

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

      6. metadata-eval 48.6%

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

      7. *-un-lft-identity 48.6%

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

      8. pow2 48.6%

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

   8. Applied egg-rr 33.9%

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

      1. unpow2 33.9%

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

      2. difference-of-squares 33.9%

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

      3. associate-+l+ 33.9%

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

      4. associate--l+ 34.6%

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

      5. associate--r- 34.6%

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

      6. +-inverses 34.6%

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

      7. associate-+l+ 34.6%

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

      8. metadata-eval 34.6%

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

      9. +-commutative 34.6%

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

      10. associate--l+ 46.7%

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

      11. associate--r- 46.7%

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

      12. +-inverses 46.7%

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

      13. associate-+l+ 46.7%

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

      14. metadata-eval 46.7%

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

      15. +-commutative 46.7%

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

   10. Simplified 46.7%

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

   if -5.3999999999999998e-211 < x < 5.8e6

   1. Initial program 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 80.3%

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

   if 5.8e6 < x < 9.20000000000000067e69

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 73.8%

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

      1. div-sub 73.8%

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

      2. mul-1-neg 73.8%

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

      3. rec-exp 73.8%

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

      4. +-inverses 73.8%

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

      5. metadata-eval 73.8%

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

   7. Simplified 73.8%

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

   if 9.20000000000000067e69 < 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 34.5%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. *-un-lft-identity 1.9%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 33.3%

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

      4. sqr-neg 33.3%

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

      5. sqrt-unprod 33.3%

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

      6. add-sqr-sqrt 33.3%

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

   10. Applied egg-rr 33.3%

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

       1. *-lft-identity 33.3%

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

   12. Simplified 33.3%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(\varepsilon + 1\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - \varepsilon\right)\right)\right)}{\varepsilon + 1}}{2}\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+69}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - eps\_m\right)} + \left(1 + x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + \frac{1}{1 + x \cdot \left(eps\_m + 1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e+27)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x -2e-295)
     (/ (+ (exp (* x (- -1.0 eps_m))) (+ 1.0 (* x eps_m))) 2.0)
     (/ (+ (exp (* x eps_m)) (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0))))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e+27) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -2e-295) {
		tmp = (exp((x * (-1.0 - eps_m))) + (1.0 + (x * eps_m))) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e+27) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -2e-295) {
		tmp = (Math.exp((x * (-1.0 - eps_m))) + (1.0 + (x * eps_m))) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4e+27:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -2e-295:
		tmp = (math.exp((x * (-1.0 - eps_m))) + (1.0 + (x * eps_m))) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e+27)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -2e-295)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + Float64(1.0 + Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e+27], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2e-295], N[(N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000001e27

   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 60.7%

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

   6. Taylor expanded in eps around 0 40.5%

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

      1. expm1-define 40.5%

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

      2. neg-mul-1 40.5%

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

   8. Simplified 40.5%

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

   if -4.0000000000000001e27 < x < -2.00000000000000012e-295

   1. Initial program 56.9%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around -inf 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-rgt-out-- 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 81.1%

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

   if -2.00000000000000012e-295 < x

   1. Initial program 73.9%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around inf 83.7%

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

      1. *-commutative 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in x around 0 60.3%

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

       1. +-commutative 60.3%

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

   11. Simplified 60.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - eps\_m\right)} + \left(1 + x \cdot eps\_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -6.5e+24)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x -1e-299)
     (/ (+ (exp (* x (- -1.0 eps_m))) (+ 1.0 (* 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 <= -6.5e+24) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -1e-299) {
		tmp = (exp((x * (-1.0 - eps_m))) + (1.0 + (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 <= -6.5e+24) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -1e-299) {
		tmp = (Math.exp((x * (-1.0 - eps_m))) + (1.0 + (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 <= -6.5e+24:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -1e-299:
		tmp = (math.exp((x * (-1.0 - eps_m))) + (1.0 + (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 <= -6.5e+24)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -1e-299)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + Float64(1.0 + 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, -6.5e+24], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1e-299], N[(N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * 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 3 regimes

2. if x < -6.4999999999999996e24

   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 60.7%

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

   6. Taylor expanded in eps around 0 40.5%

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

      1. expm1-define 40.5%

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

      2. neg-mul-1 40.5%

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

   8. Simplified 40.5%

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

   if -6.4999999999999996e24 < x < -9.99999999999999992e-300

   1. Initial program 56.9%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around -inf 100.0%

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

       1. rec-exp 100.0%

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

       2. distribute-rgt-out-- 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around 0 81.1%

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

   if -9.99999999999999992e-300 < x

   1. Initial program 73.9%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in eps around inf 98.4%

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

   6. Taylor expanded in eps around inf 83.7%

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

      1. *-commutative 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in x around 0 60.4%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+24}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 63.1%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in eps around 0 28.6%

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

      1. associate-+r+ 66.3%

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

      2. mul-1-neg 66.3%

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

      3. sub-neg 66.3%

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

      4. +-inverses 66.3%

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

      5. distribute-lft-out 66.3%

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

      6. distribute-rgt1-in 66.3%

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

      7. mul-1-neg 66.3%

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

   7. Simplified 66.3%

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

   if 1 < 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 69.5%

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

      1. +-commutative 69.5%

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

      2. mul-1-neg 69.5%

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

      3. distribute-rgt-neg-in 69.5%

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

      4. *-commutative 69.5%

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

      5. distribute-rgt-neg-in 69.5%

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

      6. mul-1-neg 69.5%

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

      7. distribute-lft-in 69.5%

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

      8. metadata-eval 69.5%

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

      9. associate-*r/ 69.5%

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

      10. metadata-eval 69.5%

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

   7. Simplified 69.5%

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

   8. Taylor expanded in eps around inf 80.9%

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

      1. associate--l+ 80.9%

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

      2. associate-*r/ 80.9%

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

      3. associate-*r* 80.9%

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

      4. remove-double-neg 80.9%

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

      5. neg-mul-1 80.9%

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

      6. sub-neg 80.9%

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

      7. associate-*r* 80.9%

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

      8. div-sub 80.9%

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

   10. Simplified 80.9%

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

4. Final simplification 70.3%

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

Alternative 8: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(eps\_m + 1\right) \cdot \left(1 + \left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - eps\_m\right)\right)\right)}{eps\_m + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -720.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x -4.8e-211)
     (/
      (+
       2.0
       (*
        x
        (/
         (* (+ eps_m 1.0) (+ 1.0 (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) eps_m))))
         (+ eps_m 1.0))))
      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 <= -720.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -4.8e-211) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 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 <= -720.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -4.8e-211) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 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 <= -720.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -4.8e-211:
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 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 <= -720.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -4.8e-211)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps_m + 1.0) * Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - eps_m)))) / Float64(eps_m + 1.0)))) / 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, -720.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.8e-211], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m + 1.0), $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 3 regimes

2. if x < -720

   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 62.7%

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

   6. Taylor expanded in eps around 0 38.5%

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

      1. expm1-define 38.5%

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

      2. neg-mul-1 38.5%

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

   8. Simplified 38.5%

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

   if -720 < x < -4.8000000000000004e-211

   1. Initial program 62.1%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in x around 0 57.4%

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

   6. Taylor expanded in eps around 0 56.5%

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

      1. associate--l+ 56.5%

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

      2. flip-+ 55.5%

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

      3. *-commutative 55.5%

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

      4. *-commutative 55.5%

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

      5. swap-sqr 55.5%

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

      6. metadata-eval 55.5%

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

      7. *-un-lft-identity 55.5%

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

      8. pow2 55.5%

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

   8. Applied egg-rr 36.0%

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

      1. unpow2 36.0%

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

      2. difference-of-squares 36.0%

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

      3. associate-+l+ 36.0%

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

      4. associate--l+ 36.9%

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

      5. associate--r- 36.9%

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

      6. +-inverses 36.9%

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

      7. associate-+l+ 36.9%

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

      8. metadata-eval 36.9%

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

      9. +-commutative 36.9%

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

      10. associate--l+ 52.9%

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

      11. associate--r- 52.9%

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

      12. +-inverses 52.9%

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

      13. associate-+l+ 52.9%

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

      14. metadata-eval 52.9%

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

      15. +-commutative 52.9%

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

   10. Simplified 52.9%

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

   if -4.8000000000000004e-211 < x

   1. Initial program 70.1%

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

   3. Simplified 65.9%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in eps around inf 85.5%

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

      1. *-commutative 85.5%

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

   8. Simplified 85.5%

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

   9. Taylor expanded in x around 0 64.8%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(\varepsilon + 1\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - \varepsilon\right)\right)\right)}{\varepsilon + 1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(eps\_m + 1\right) \cdot \left(1 + \left(\frac{1}{eps\_m} + \left(\frac{1}{eps\_m} - eps\_m\right)\right)\right)}{eps\_m + 1}}{2}\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+71}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (/
           (*
            x
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
           eps_m)
          2.0)))
   (if (<= x -2.6e+77)
     t_0
     (if (<= x -4.5e-212)
       (/
        (+
         2.0
         (*
          x
          (/
           (* (+ eps_m 1.0) (+ 1.0 (+ (/ 1.0 eps_m) (- (/ 1.0 eps_m) eps_m))))
           (+ eps_m 1.0))))
        2.0)
       (if (<= x 5800000.0) 1.0 (if (<= x 8.4e+71) 0.0 t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = t_0;
	} else if (x <= -4.5e-212) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 8.4e+71) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    if (x <= (-2.6d+77)) then
        tmp = t_0
    else if (x <= (-4.5d-212)) then
        tmp = (2.0d0 + (x * (((eps_m + 1.0d0) * (1.0d0 + ((1.0d0 / eps_m) + ((1.0d0 / eps_m) - eps_m)))) / (eps_m + 1.0d0)))) / 2.0d0
    else if (x <= 5800000.0d0) then
        tmp = 1.0d0
    else if (x <= 8.4d+71) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = t_0;
	} else if (x <= -4.5e-212) {
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 8.4e+71) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	tmp = 0
	if x <= -2.6e+77:
		tmp = t_0
	elif x <= -4.5e-212:
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0
	elif x <= 5800000.0:
		tmp = 1.0
	elif x <= 8.4e+71:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = t_0;
	elseif (x <= -4.5e-212)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(eps_m + 1.0) * Float64(1.0 + Float64(Float64(1.0 / eps_m) + Float64(Float64(1.0 / eps_m) - eps_m)))) / Float64(eps_m + 1.0)))) / 2.0);
	elseif (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 8.4e+71)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	tmp = 0.0;
	if (x <= -2.6e+77)
		tmp = t_0;
	elseif (x <= -4.5e-212)
		tmp = (2.0 + (x * (((eps_m + 1.0) * (1.0 + ((1.0 / eps_m) + ((1.0 / eps_m) - eps_m)))) / (eps_m + 1.0)))) / 2.0;
	elseif (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 8.4e+71)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.6e+77], t$95$0, If[LessEqual[x, -4.5e-212], N[(N[(2.0 + N[(x * N[(N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(1.0 + N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5800000.0], 1.0, If[LessEqual[x, 8.4e+71], 0.0, t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6000000000000002e77 or 8.39999999999999957e71 < 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 43.7%

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

   6. Taylor expanded in eps around 0 13.2%

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

      1. expm1-define 13.2%

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

      2. neg-mul-1 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in x around 0 34.6%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -2.6000000000000002e77 < x < -4.4999999999999999e-212

   1. Initial program 71.4%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around 0 43.3%

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

      1. associate--l+ 43.3%

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

      2. flip-+ 48.6%

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

      3. *-commutative 48.6%

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

      4. *-commutative 48.6%

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

      5. swap-sqr 48.6%

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

      6. metadata-eval 48.6%

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

      7. *-un-lft-identity 48.6%

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

      8. pow2 48.6%

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

   8. Applied egg-rr 33.9%

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

      1. unpow2 33.9%

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

      2. difference-of-squares 33.9%

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

      3. associate-+l+ 33.9%

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

      4. associate--l+ 34.6%

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

      5. associate--r- 34.6%

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

      6. +-inverses 34.6%

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

      7. associate-+l+ 34.6%

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

      8. metadata-eval 34.6%

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

      9. +-commutative 34.6%

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

      10. associate--l+ 46.7%

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

      11. associate--r- 46.7%

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

      12. +-inverses 46.7%

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

      13. associate-+l+ 46.7%

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

      14. metadata-eval 46.7%

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

      15. +-commutative 46.7%

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

   10. Simplified 46.7%

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

   if -4.4999999999999999e-212 < x < 5.8e6

   1. Initial program 50.8%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in x around 0 80.3%

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

   if 5.8e6 < x < 8.39999999999999957e71

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 73.8%

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

      1. div-sub 73.8%

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

      2. mul-1-neg 73.8%

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

      3. rec-exp 73.8%

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

      4. +-inverses 73.8%

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

      5. metadata-eval 73.8%

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

   7. Simplified 73.8%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-212}:\\ \;\;\;\;\frac{2 + x \cdot \frac{\left(\varepsilon + 1\right) \cdot \left(1 + \left(\frac{1}{\varepsilon} + \left(\frac{1}{\varepsilon} - \varepsilon\right)\right)\right)}{\varepsilon + 1}}{2}\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+71}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 210:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+71}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (/
           (*
            x
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
           eps_m)
          2.0)))
   (if (<= x -2.6e+77)
     t_0
     (if (<= x 210.0)
       (/ (- 2.0 (* x eps_m)) 2.0)
       (if (<= x 8.4e+71) 0.0 t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = t_0;
	} else if (x <= 210.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 8.4e+71) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    if (x <= (-2.6d+77)) then
        tmp = t_0
    else if (x <= 210.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= 8.4d+71) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = t_0;
	} else if (x <= 210.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 8.4e+71) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	tmp = 0
	if x <= -2.6e+77:
		tmp = t_0
	elif x <= 210.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= 8.4e+71:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = t_0;
	elseif (x <= 210.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= 8.4e+71)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	tmp = 0.0;
	if (x <= -2.6e+77)
		tmp = t_0;
	elseif (x <= 210.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= 8.4e+71)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.6e+77], t$95$0, If[LessEqual[x, 210.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.4e+71], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000002e77 or 8.39999999999999957e71 < 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 43.7%

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

   6. Taylor expanded in eps around 0 13.2%

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

      1. expm1-define 13.2%

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

      2. neg-mul-1 13.2%

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

   8. Simplified 13.2%

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

   9. Taylor expanded in x around 0 34.6%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -2.6000000000000002e77 < x < 210

   1. Initial program 57.9%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in x around 0 67.7%

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

   6. Taylor expanded in eps around 0 66.9%

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

   7. Taylor expanded in eps around inf 66.9%

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

      1. mul-1-neg 66.9%

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

      2. distribute-lft-neg-out 66.9%

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

      3. *-commutative 66.9%

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

   9. Simplified 66.9%

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

   if 210 < x < 8.39999999999999957e71

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 65.3%

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

      1. div-sub 65.3%

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

      2. mul-1-neg 65.3%

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

      3. rec-exp 65.3%

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

      4. +-inverses 65.3%

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

      5. metadata-eval 65.3%

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

   7. Simplified 65.3%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 210:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+71}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.3% accurate, 10.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot 0.5\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 210.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (<= x 2.2e+150) 0.0 (/ (/ (* x (+ -1.0 (* x 0.5))) eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 2.2e+150) {
		tmp = 0.0;
	} else {
		tmp = ((x * (-1.0 + (x * 0.5))) / 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 <= 210.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= 2.2d+150) then
        tmp = 0.0d0
    else
        tmp = ((x * ((-1.0d0) + (x * 0.5d0))) / 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 <= 210.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 2.2e+150) {
		tmp = 0.0;
	} else {
		tmp = ((x * (-1.0 + (x * 0.5))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 210.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= 2.2e+150:
		tmp = 0.0
	else:
		tmp = ((x * (-1.0 + (x * 0.5))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 210.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= 2.2e+150)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * 0.5))) / 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 <= 210.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= 2.2e+150)
		tmp = 0.0;
	else
		tmp = ((x * (-1.0 + (x * 0.5))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 210.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.2e+150], 0.0, N[(N[(N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 210

   1. Initial program 63.3%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in x around 0 59.4%

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

   6. Taylor expanded in eps around 0 62.6%

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

   7. Taylor expanded in eps around inf 62.6%

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

      1. mul-1-neg 62.6%

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

      2. distribute-lft-neg-out 62.6%

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

      3. *-commutative 62.6%

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

   9. Simplified 62.6%

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

   if 210 < x < 2.19999999999999999e150

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 55.0%

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

      1. div-sub 55.0%

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

      2. mul-1-neg 55.0%

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

      3. rec-exp 55.0%

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

      4. +-inverses 55.0%

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

      5. metadata-eval 55.0%

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

   7. Simplified 55.0%

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

   if 2.19999999999999999e150 < 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 39.9%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 36.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+150}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot 0.5\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;\left(x \cdot eps\_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+218}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.85e-13)
   (* (* x eps_m) -0.5)
   (if (<= x 5800000.0) 1.0 (if (<= x 1e+218) 0.0 (* eps_m (* x 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.85e-13) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 1e+218) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.85d-13)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 5800000.0d0) then
        tmp = 1.0d0
    else if (x <= 1d+218) then
        tmp = 0.0d0
    else
        tmp = eps_m * (x * 0.5d0)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.85e-13) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 1e+218) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.85e-13:
		tmp = (x * eps_m) * -0.5
	elif x <= 5800000.0:
		tmp = 1.0
	elif x <= 1e+218:
		tmp = 0.0
	else:
		tmp = eps_m * (x * 0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.85e-13)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 1e+218)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.85e-13)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 1e+218)
		tmp = 0.0;
	else
		tmp = eps_m * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.85e-13], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 5800000.0], 1.0, If[LessEqual[x, 1e+218], 0.0, N[(eps$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.84999999999999994e-13

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in eps around 0 20.4%

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

   7. Taylor expanded in x around 0 20.4%

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

      1. neg-mul-1 20.4%

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

      2. distribute-neg-in 20.4%

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

      3. metadata-eval 20.4%

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

      4. sub-neg 20.4%

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

      5. associate-+l- 20.4%

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

      6. +-inverses 20.4%

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

      7. metadata-eval 20.4%

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

   9. Simplified 20.4%

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

   10. Taylor expanded in eps around inf 20.4%

       \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   11. Step-by-step derivation

       1. *-commutative 20.4%

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

       2. *-commutative 20.4%

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

   12. Simplified 20.4%

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

   if -1.84999999999999994e-13 < x < 5.8e6

   1. Initial program 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in x around 0 74.7%

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

   if 5.8e6 < x < 1.00000000000000008e218

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 52.8%

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

      1. div-sub 52.8%

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

      2. mul-1-neg 52.8%

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

      3. rec-exp 52.8%

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

      4. +-inverses 52.8%

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

      5. metadata-eval 52.8%

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

   7. Simplified 52.8%

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

   if 1.00000000000000008e218 < 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 43.6%

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

   6. Taylor expanded in x around 0 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-in 37.7%

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

      3. *-commutative 37.7%

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

      4. distribute-rgt-neg-in 37.7%

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

      5. mul-1-neg 37.7%

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

      6. associate-*r* 37.7%

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

      7. *-commutative 37.7%

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

      8. associate-*l* 37.7%

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

      9. distribute-lft-in 37.7%

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

      10. metadata-eval 37.7%

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

      11. neg-mul-1 37.7%

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

      12. distribute-neg-frac 37.7%

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

      13. metadata-eval 37.7%

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

   8. Simplified 37.7%

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

   9. Taylor expanded in eps around inf 38.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 38.0%

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

       2. *-commutative 38.0%

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

       3. *-commutative 38.0%

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

       4. associate-*r* 38.0%

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

   11. Simplified 38.0%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+218}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 245:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+216}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 245.0)
   (/ (- 2.0 (* x eps_m)) 2.0)
   (if (<= x 3.3e+216) 0.0 (* eps_m (* x 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 245.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 3.3e+216) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 245.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else if (x <= 3.3d+216) then
        tmp = 0.0d0
    else
        tmp = eps_m * (x * 0.5d0)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 245.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else if (x <= 3.3e+216) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 245.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	elif x <= 3.3e+216:
		tmp = 0.0
	else:
		tmp = eps_m * (x * 0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 245.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	elseif (x <= 3.3e+216)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 245.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	elseif (x <= 3.3e+216)
		tmp = 0.0;
	else
		tmp = eps_m * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 245.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.3e+216], 0.0, N[(eps$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 245

   1. Initial program 63.3%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in x around 0 59.4%

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

   6. Taylor expanded in eps around 0 62.6%

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

   7. Taylor expanded in eps around inf 62.6%

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

      1. mul-1-neg 62.6%

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

      2. distribute-lft-neg-out 62.6%

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

      3. *-commutative 62.6%

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

   9. Simplified 62.6%

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

   if 245 < x < 3.3e216

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 50.8%

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

      1. div-sub 50.8%

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

      2. mul-1-neg 50.8%

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

      3. rec-exp 50.8%

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

      4. +-inverses 50.8%

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

      5. metadata-eval 50.8%

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

   7. Simplified 50.8%

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

   if 3.3e216 < 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 43.6%

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

   6. Taylor expanded in x around 0 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-in 37.7%

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

      3. *-commutative 37.7%

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

      4. distribute-rgt-neg-in 37.7%

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

      5. mul-1-neg 37.7%

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

      6. associate-*r* 37.7%

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

      7. *-commutative 37.7%

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

      8. associate-*l* 37.7%

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

      9. distribute-lft-in 37.7%

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

      10. metadata-eval 37.7%

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

      11. neg-mul-1 37.7%

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

      12. distribute-neg-frac 37.7%

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

      13. metadata-eval 37.7%

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

   8. Simplified 37.7%

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

   9. Taylor expanded in eps around inf 38.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 38.0%

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

       2. *-commutative 38.0%

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

       3. *-commutative 38.0%

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

       4. associate-*r* 38.0%

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

   11. Simplified 38.0%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 245:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+216}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;eps\_m \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 5800000.0) 1.0 (if (<= x 6.6e+215) 0.0 (* eps_m (* x 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 6.6e+215) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 5800000.0d0) then
        tmp = 1.0d0
    else if (x <= 6.6d+215) then
        tmp = 0.0d0
    else
        tmp = eps_m * (x * 0.5d0)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 5800000.0) {
		tmp = 1.0;
	} else if (x <= 6.6e+215) {
		tmp = 0.0;
	} else {
		tmp = eps_m * (x * 0.5);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 5800000.0:
		tmp = 1.0
	elif x <= 6.6e+215:
		tmp = 0.0
	else:
		tmp = eps_m * (x * 0.5)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 6.6e+215)
		tmp = 0.0;
	else
		tmp = Float64(eps_m * Float64(x * 0.5));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 5800000.0)
		tmp = 1.0;
	elseif (x <= 6.6e+215)
		tmp = 0.0;
	else
		tmp = eps_m * (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 5800000.0], 1.0, If[LessEqual[x, 6.6e+215], 0.0, N[(eps$95$m * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.8e6

   1. Initial program 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around 0 58.8%

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

   if 5.8e6 < x < 6.5999999999999997e215

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 52.8%

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

      1. div-sub 52.8%

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

      2. mul-1-neg 52.8%

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

      3. rec-exp 52.8%

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

      4. +-inverses 52.8%

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

      5. metadata-eval 52.8%

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

   7. Simplified 52.8%

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

   if 6.5999999999999997e215 < 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 43.6%

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

   6. Taylor expanded in x around 0 37.7%

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

      1. mul-1-neg 37.7%

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

      2. distribute-rgt-neg-in 37.7%

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

      3. *-commutative 37.7%

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

      4. distribute-rgt-neg-in 37.7%

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

      5. mul-1-neg 37.7%

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

      6. associate-*r* 37.7%

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

      7. *-commutative 37.7%

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

      8. associate-*l* 37.7%

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

      9. distribute-lft-in 37.7%

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

      10. metadata-eval 37.7%

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

      11. neg-mul-1 37.7%

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

      12. distribute-neg-frac 37.7%

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

      13. metadata-eval 37.7%

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

   8. Simplified 37.7%

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

   9. Taylor expanded in eps around inf 38.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 38.0%

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

       2. *-commutative 38.0%

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

       3. *-commutative 38.0%

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

       4. associate-*r* 38.0%

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

   11. Simplified 38.0%

       \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+215}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.9% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 5800000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 5800000.0) {
		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 <= 5800000.0d0) 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 <= 5800000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 5800000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 5800000.0)
		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 <= 5800000.0)
		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, 5800000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e6

   1. Initial program 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in x around 0 58.8%

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

   if 5.8e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.1%

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

      1. div-sub 47.1%

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

      2. mul-1-neg 47.1%

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

      3. rec-exp 47.1%

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

      4. +-inverses 47.1%

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

      5. metadata-eval 47.1%

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

   7. Simplified 47.1%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5800000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 15.8% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.2%

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

3. Simplified 67.2%

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

5. Taylor expanded in eps around 0 13.9%

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

   1. div-sub 13.9%

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

   2. mul-1-neg 13.9%

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

   3. rec-exp 13.9%

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

   4. +-inverses 14.2%

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

   5. metadata-eval 14.2%

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

7. Simplified 14.2%

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

Reproduce

herbie shell --seed 2024188 
(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))