NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 98.8%

Time: 13.2s

Alternatives: 17

Speedup: 1.9×

• 37.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% 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: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 65.0%

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

5. Taylor expanded in eps around inf 100.0%

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

   1. mul-1-neg 100.0%

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

   2. exp-neg 100.0%

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

   3. exp-prod 92.6%

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

   4. add-sqr-sqrt 53.6%

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

   5. sqrt-unprod 83.7%

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

   6. sqr-neg 83.7%

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

   7. sqrt-unprod 30.2%

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

   8. add-sqr-sqrt 60.7%

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

   9. neg-mul-1 60.7%

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

   10. exp-to-pow 60.7%

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

   11. add-log-exp 60.6%

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

   12. neg-mul-1 60.6%

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

   13. add-sqr-sqrt 30.1%

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

   14. sqrt-unprod 87.5%

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

   15. sqr-neg 87.5%

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

   16. sqrt-unprod 58.6%

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

   17. add-sqr-sqrt 100.0%

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

   18. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 85.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \varepsilon}\\ \mathbf{if}\;x \leq 1.7 \cdot 10^{+202}:\\ \;\;\;\;\frac{t\_0 + \frac{1}{t\_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x eps))))
   (if (<= x 1.7e+202) (/ (+ t_0 (/ 1.0 t_0)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double t_0 = exp((x * eps));
	double tmp;
	if (x <= 1.7e+202) {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * eps))
    if (x <= 1.7d+202) then
        tmp = (t_0 + (1.0d0 / t_0)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * eps));
	double tmp;
	if (x <= 1.7e+202) {
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * eps))
	tmp = 0
	if x <= 1.7e+202:
		tmp = (t_0 + (1.0 / t_0)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * eps))
	tmp = 0.0
	if (x <= 1.7e+202)
		tmp = Float64(Float64(t_0 + Float64(1.0 / t_0)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * eps));
	tmp = 0.0;
	if (x <= 1.7e+202)
		tmp = (t_0 + (1.0 / t_0)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.7e+202], N[(N[(t$95$0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e202

   1. Initial program 75.8%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. exp-prod 92.0%

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

      4. add-sqr-sqrt 49.6%

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

      5. sqrt-unprod 82.4%

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

      6. sqr-neg 82.4%

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

      7. sqrt-unprod 32.7%

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

      8. add-sqr-sqrt 64.9%

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

      9. neg-mul-1 64.9%

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

      10. exp-to-pow 64.9%

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

      11. add-log-exp 64.7%

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

      12. neg-mul-1 64.7%

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

      13. add-sqr-sqrt 32.7%

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

      14. sqrt-unprod 86.4%

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

      15. sqr-neg 86.4%

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

      16. sqrt-unprod 55.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in eps around inf 93.4%

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

      1. *-commutative 93.4%

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

   10. Simplified 93.4%

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

   11. Taylor expanded in eps around inf 91.7%

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

       1. *-commutative 91.7%

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

   13. Simplified 91.7%

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

   if 1.7e202 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

   7. Taylor expanded in eps around 0 70.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{+202}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + \frac{1}{e^{x \cdot \varepsilon}}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 65.0%

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

5. Taylor expanded in eps around inf 100.0%

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

6. Taylor expanded in eps around 0 100.0%

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

   1. associate-*r* 100.0%

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

   2. distribute-rgt-in 100.0%

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

   3. mul-1-neg 100.0%

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

   4. unsub-neg 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 4: 69.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{\frac{0.5}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 - \varepsilon \cdot \varepsilon}{1 + \varepsilon} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35)
   (/ (/ 0.5 (exp x)) 2.0)
   (if (<= x 2.4e-75)
     1.0
     (if (<= x 720.0)
       (/
        (+
         2.0
         (* x (* (/ (- 1.0 (* eps eps)) (+ 1.0 eps)) (+ -1.0 (/ -1.0 eps)))))
        2.0)
       (if (<= x 4e+202) (/ (expm1 x) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35) {
		tmp = (0.5 / exp(x)) / 2.0;
	} else if (x <= 2.4e-75) {
		tmp = 1.0;
	} else if (x <= 720.0) {
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else if (x <= 4e+202) {
		tmp = expm1(x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.35) {
		tmp = (0.5 / Math.exp(x)) / 2.0;
	} else if (x <= 2.4e-75) {
		tmp = 1.0;
	} else if (x <= 720.0) {
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else if (x <= 4e+202) {
		tmp = Math.expm1(x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35:
		tmp = (0.5 / math.exp(x)) / 2.0
	elif x <= 2.4e-75:
		tmp = 1.0
	elif x <= 720.0:
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0
	elif x <= 4e+202:
		tmp = math.expm1(x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(Float64(0.5 / exp(x)) / 2.0);
	elseif (x <= 2.4e-75)
		tmp = 1.0;
	elseif (x <= 720.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - Float64(eps * eps)) / Float64(1.0 + eps)) * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	elseif (x <= 4e+202)
		tmp = Float64(expm1(x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35], N[(N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.4e-75], 1.0, If[LessEqual[x, 720.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+202], N[(N[(Exp[x] - 1), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.3500000000000001

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 51.9%

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

      6. sqr-neg 51.9%

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

      7. sqrt-unprod 51.9%

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

      8. add-sqr-sqrt 51.9%

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

      9. neg-mul-1 51.9%

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

      10. exp-to-pow 51.9%

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

      11. add-log-exp 51.9%

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

      12. neg-mul-1 51.9%

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

      13. add-sqr-sqrt 51.9%

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

      14. sqrt-unprod 51.9%

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

      15. sqr-neg 51.9%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

       2. count-2 100.0%

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

       3. associate-/r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{e^{x}}}{2} \]

   14. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{e^{x}}}}{2} \]

   if -1.3500000000000001 < x < 2.40000000000000019e-75

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in x around 0 82.5%

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

   if 2.40000000000000019e-75 < x < 720

   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 73.9%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in x around 0 21.4%

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

      1. mul-1-neg 21.4%

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

      2. distribute-rgt-neg-in 21.4%

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

      3. *-commutative 21.4%

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

      4. distribute-rgt-neg-in 21.4%

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

      5. distribute-neg-in 21.4%

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

      6. metadata-eval 21.4%

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

      7. distribute-neg-frac 21.4%

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

      8. metadata-eval 21.4%

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

   8. Simplified 21.4%

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

      1. sub-neg 21.4%

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

      2. flip-+ 39.5%

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

      3. metadata-eval 39.5%

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

   10. Applied egg-rr 39.5%

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

   if 720 < x < 3.9999999999999996e202

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 29.5%

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

   7. Taylor expanded in eps around inf 29.4%

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

      1. *-commutative 71.7%

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

   9. Simplified 29.4%

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

   11. Applied egg-rr 0.0%

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

       1. +-commutative 0.0%

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

       2. associate--l+ 0.0%

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

       3. +-inverses 64.2%

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

       4. metadata-eval 64.2%

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

       5. metadata-eval 64.2%

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

       6. *-rgt-identity 64.2%

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

       7. metadata-eval 64.2%

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

       8. sub-neg 64.2%

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

       9. expm1-undefine 64.2%

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

   13. Simplified 64.2%

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

   if 3.9999999999999996e202 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

   7. Taylor expanded in eps around 0 70.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{\frac{0.5}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 - \varepsilon \cdot \varepsilon}{1 + \varepsilon} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 41:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 - \varepsilon \cdot \varepsilon}{1 + \varepsilon} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-11)
   (/ (* eps (* (+ x -8.0) (+ -8.0 (* x 2.0)))) 2.0)
   (if (<= x 2.15e-75)
     1.0
     (if (<= x 41.0)
       (/
        (+
         2.0
         (* x (* (/ (- 1.0 (* eps eps)) (+ 1.0 eps)) (+ -1.0 (/ -1.0 eps)))))
        2.0)
       (if (<= x 1.65e+205) (/ (expm1 x) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 2.15e-75) {
		tmp = 1.0;
	} else if (x <= 41.0) {
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else if (x <= 1.65e+205) {
		tmp = expm1(x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 2.15e-75) {
		tmp = 1.0;
	} else if (x <= 41.0) {
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else if (x <= 1.65e+205) {
		tmp = Math.expm1(x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-11:
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0
	elif x <= 2.15e-75:
		tmp = 1.0
	elif x <= 41.0:
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0
	elif x <= 1.65e+205:
		tmp = math.expm1(x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(-8.0 + Float64(x * 2.0)))) / 2.0);
	elseif (x <= 2.15e-75)
		tmp = 1.0;
	elseif (x <= 41.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - Float64(eps * eps)) / Float64(1.0 + eps)) * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	elseif (x <= 1.65e+205)
		tmp = Float64(expm1(x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-11], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(-8.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.15e-75], 1.0, If[LessEqual[x, 41.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e+205], N[(N[(Exp[x] - 1), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.70000000000000005e-11

   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 51.6%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. distribute-rgt-neg-in 23.6%

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

      3. *-commutative 23.6%

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

      4. distribute-rgt-neg-in 23.6%

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

      5. distribute-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

      7. distribute-neg-frac 23.6%

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

      8. metadata-eval 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in eps around inf 23.6%

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

       1. associate-*r/ 23.6%

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

       2. metadata-eval 23.6%

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

   11. Simplified 23.6%

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

   13. Applied egg-rr 39.9%

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

       1. *-commutative 39.9%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + \left(x + -8\right)\right)\right)}}{2} \]

       2. associate-+r+ 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \color{blue}{\left(\left(x + x\right) + -8\right)}\right)}{2} \]

       3. count-2 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(\color{blue}{2 \cdot x} + -8\right)\right)}{2} \]

   15. Simplified 39.9%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(2 \cdot x + -8\right)\right)}}{2} \]

   if -2.70000000000000005e-11 < x < 2.15e-75

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in x around 0 83.2%

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

   if 2.15e-75 < x < 41

   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 73.9%

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

   5. Taylor expanded in x around 0 40.7%

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

   6. Taylor expanded in x around 0 21.4%

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

      1. mul-1-neg 21.4%

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

      2. distribute-rgt-neg-in 21.4%

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

      3. *-commutative 21.4%

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

      4. distribute-rgt-neg-in 21.4%

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

      5. distribute-neg-in 21.4%

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

      6. metadata-eval 21.4%

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

      7. distribute-neg-frac 21.4%

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

      8. metadata-eval 21.4%

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

   8. Simplified 21.4%

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

      1. sub-neg 21.4%

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

      2. flip-+ 39.5%

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

      3. metadata-eval 39.5%

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

   10. Applied egg-rr 39.5%

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

   if 41 < x < 1.6500000000000001e205

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 29.5%

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

   7. Taylor expanded in eps around inf 29.4%

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

      1. *-commutative 71.7%

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

   9. Simplified 29.4%

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

   11. Applied egg-rr 0.0%

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

       1. +-commutative 0.0%

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

       2. associate--l+ 0.0%

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

       3. +-inverses 64.2%

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

       4. metadata-eval 64.2%

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

       5. metadata-eval 64.2%

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

       6. *-rgt-identity 64.2%

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

       7. metadata-eval 64.2%

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

       8. sub-neg 64.2%

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

       9. expm1-undefine 64.2%

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

   13. Simplified 64.2%

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

   if 1.6500000000000001e205 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

   7. Taylor expanded in eps around 0 70.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 41:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 - \varepsilon \cdot \varepsilon}{1 + \varepsilon} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-245}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 - x\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.8e-245)
   (/ (+ (exp (* x (- -1.0 eps))) (- 1.0 x)) 2.0)
   (if (<= x 1.65e+205) (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.8e-245) {
		tmp = (exp((x * (-1.0 - eps))) + (1.0 - x)) / 2.0;
	} else if (x <= 1.65e+205) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.8d-245)) then
        tmp = (exp((x * ((-1.0d0) - eps))) + (1.0d0 - x)) / 2.0d0
    else if (x <= 1.65d+205) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.8e-245) {
		tmp = (Math.exp((x * (-1.0 - eps))) + (1.0 - x)) / 2.0;
	} else if (x <= 1.65e+205) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.8e-245:
		tmp = (math.exp((x * (-1.0 - eps))) + (1.0 - x)) / 2.0
	elif x <= 1.65e+205:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.8e-245)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + Float64(1.0 - x)) / 2.0);
	elseif (x <= 1.65e+205)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.8e-245)
		tmp = (exp((x * (-1.0 - eps))) + (1.0 - x)) / 2.0;
	elseif (x <= 1.65e+205)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.8e-245], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e+205], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999999e-245

   1. Initial program 75.5%

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

   3. Simplified 62.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 66.6%

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

   7. Taylor expanded in eps around 0 70.5%

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

      1. neg-mul-1 70.5%

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

      2. unsub-neg 70.5%

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

   9. Simplified 70.5%

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

   if -5.7999999999999999e-245 < x < 1.6500000000000001e205

   1. Initial program 76.0%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in x around 0 62.0%

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

   if 1.6500000000000001e205 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

   7. Taylor expanded in eps around 0 70.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-245}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 - x\right)}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{\frac{0.5}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+205}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.55)
   (/ (/ 0.5 (exp x)) 2.0)
   (if (<= x 1.05e+205) (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.55) {
		tmp = (0.5 / exp(x)) / 2.0;
	} else if (x <= 1.05e+205) {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.55d0)) then
        tmp = (0.5d0 / exp(x)) / 2.0d0
    else if (x <= 1.05d+205) then
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.55) {
		tmp = (0.5 / Math.exp(x)) / 2.0;
	} else if (x <= 1.05e+205) {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.55:
		tmp = (0.5 / math.exp(x)) / 2.0
	elif x <= 1.05e+205:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.55)
		tmp = Float64(Float64(0.5 / exp(x)) / 2.0);
	elseif (x <= 1.05e+205)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.55)
		tmp = (0.5 / exp(x)) / 2.0;
	elseif (x <= 1.05e+205)
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.55], N[(N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.05e+205], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.55000000000000004

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 51.9%

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

      6. sqr-neg 51.9%

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

      7. sqrt-unprod 51.9%

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

      8. add-sqr-sqrt 51.9%

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

      9. neg-mul-1 51.9%

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

      10. exp-to-pow 51.9%

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

      11. add-log-exp 51.9%

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

      12. neg-mul-1 51.9%

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

      13. add-sqr-sqrt 51.9%

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

      14. sqrt-unprod 51.9%

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

      15. sqr-neg 51.9%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

       2. count-2 100.0%

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

       3. associate-/r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{e^{x}}}{2} \]

   14. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{e^{x}}}}{2} \]

   if -0.55000000000000004 < x < 1.05e205

   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 53.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in x around 0 68.3%

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

   if 1.05e205 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

   7. Taylor expanded in eps around 0 70.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.55:\\ \;\;\;\;\frac{\frac{0.5}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+205}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.6:\\ \;\;\;\;\frac{\frac{0.5}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.6)
   (/ (/ 0.5 (exp x)) 2.0)
   (if (<= x 1.6e+205) (/ (+ 1.0 (exp (* x eps))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.6) {
		tmp = (0.5 / exp(x)) / 2.0;
	} else if (x <= 1.6e+205) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.6d0)) then
        tmp = (0.5d0 / exp(x)) / 2.0d0
    else if (x <= 1.6d+205) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.6) {
		tmp = (0.5 / Math.exp(x)) / 2.0;
	} else if (x <= 1.6e+205) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.6:
		tmp = (0.5 / math.exp(x)) / 2.0
	elif x <= 1.6e+205:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.6)
		tmp = Float64(Float64(0.5 / exp(x)) / 2.0);
	elseif (x <= 1.6e+205)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.6)
		tmp = (0.5 / exp(x)) / 2.0;
	elseif (x <= 1.6e+205)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.6], N[(N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+205], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.599999999999999978

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. exp-prod 100.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 51.9%

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

      6. sqr-neg 51.9%

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

      7. sqrt-unprod 51.9%

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

      8. add-sqr-sqrt 51.9%

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

      9. neg-mul-1 51.9%

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

      10. exp-to-pow 51.9%

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

      11. add-log-exp 51.9%

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

      12. neg-mul-1 51.9%

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

      13. add-sqr-sqrt 51.9%

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

      14. sqrt-unprod 51.9%

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

      15. sqr-neg 51.9%

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

      16. sqrt-unprod 0.0%

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

      17. add-sqr-sqrt 100.0%

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

      18. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   12. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

       2. count-2 100.0%

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

       3. associate-/r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \frac{\frac{\color{blue}{0.5}}{e^{x}}}{2} \]

   14. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{0.5}{e^{x}}}}{2} \]

   if -0.599999999999999978 < x < 1.59999999999999998e205

   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 53.1%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in x around 0 68.3%

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

   7. Taylor expanded in eps around inf 68.3%

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

      1. *-commutative 91.8%

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

   9. Simplified 68.3%

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

   if 1.59999999999999998e205 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

   7. Taylor expanded in eps around 0 70.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.6:\\ \;\;\;\;\frac{\frac{0.5}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+205}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+45}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 - \varepsilon \cdot \varepsilon}{1 + \varepsilon} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+117}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-11)
   (/ (* eps (* (+ x -8.0) (+ -8.0 (* x 2.0)))) 2.0)
   (if (<= x 3.5e-75)
     1.0
     (if (<= x 1.3e+45)
       (/
        (+
         2.0
         (* x (* (/ (- 1.0 (* eps eps)) (+ 1.0 eps)) (+ -1.0 (/ -1.0 eps)))))
        2.0)
       (if (<= x 1.4e+117)
         0.0
         (if (<= x 1.4e+202)
           (/ (* eps (* (+ x -8.0) (+ x -8.0))) 2.0)
           0.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 3.5e-75) {
		tmp = 1.0;
	} else if (x <= 1.3e+45) {
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else if (x <= 1.4e+117) {
		tmp = 0.0;
	} else if (x <= 1.4e+202) {
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-11)) then
        tmp = (eps * ((x + (-8.0d0)) * ((-8.0d0) + (x * 2.0d0)))) / 2.0d0
    else if (x <= 3.5d-75) then
        tmp = 1.0d0
    else if (x <= 1.3d+45) then
        tmp = (2.0d0 + (x * (((1.0d0 - (eps * eps)) / (1.0d0 + eps)) * ((-1.0d0) + ((-1.0d0) / eps))))) / 2.0d0
    else if (x <= 1.4d+117) then
        tmp = 0.0d0
    else if (x <= 1.4d+202) then
        tmp = (eps * ((x + (-8.0d0)) * (x + (-8.0d0)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 3.5e-75) {
		tmp = 1.0;
	} else if (x <= 1.3e+45) {
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	} else if (x <= 1.4e+117) {
		tmp = 0.0;
	} else if (x <= 1.4e+202) {
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-11:
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0
	elif x <= 3.5e-75:
		tmp = 1.0
	elif x <= 1.3e+45:
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0
	elif x <= 1.4e+117:
		tmp = 0.0
	elif x <= 1.4e+202:
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(-8.0 + Float64(x * 2.0)))) / 2.0);
	elseif (x <= 3.5e-75)
		tmp = 1.0;
	elseif (x <= 1.3e+45)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 - Float64(eps * eps)) / Float64(1.0 + eps)) * Float64(-1.0 + Float64(-1.0 / eps))))) / 2.0);
	elseif (x <= 1.4e+117)
		tmp = 0.0;
	elseif (x <= 1.4e+202)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(x + -8.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-11)
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	elseif (x <= 3.5e-75)
		tmp = 1.0;
	elseif (x <= 1.3e+45)
		tmp = (2.0 + (x * (((1.0 - (eps * eps)) / (1.0 + eps)) * (-1.0 + (-1.0 / eps))))) / 2.0;
	elseif (x <= 1.4e+117)
		tmp = 0.0;
	elseif (x <= 1.4e+202)
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-11], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(-8.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.5e-75], 1.0, If[LessEqual[x, 1.3e+45], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(1.0 + eps), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4e+117], 0.0, If[LessEqual[x, 1.4e+202], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(x + -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.70000000000000005e-11

   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 51.6%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. distribute-rgt-neg-in 23.6%

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

      3. *-commutative 23.6%

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

      4. distribute-rgt-neg-in 23.6%

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

      5. distribute-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

      7. distribute-neg-frac 23.6%

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

      8. metadata-eval 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in eps around inf 23.6%

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

       1. associate-*r/ 23.6%

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

       2. metadata-eval 23.6%

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

   11. Simplified 23.6%

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

   13. Applied egg-rr 39.9%

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

       1. *-commutative 39.9%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + \left(x + -8\right)\right)\right)}}{2} \]

       2. associate-+r+ 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \color{blue}{\left(\left(x + x\right) + -8\right)}\right)}{2} \]

       3. count-2 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(\color{blue}{2 \cdot x} + -8\right)\right)}{2} \]

   15. Simplified 39.9%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(2 \cdot x + -8\right)\right)}}{2} \]

   if -2.70000000000000005e-11 < x < 3.49999999999999985e-75

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in x around 0 83.2%

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

   if 3.49999999999999985e-75 < x < 1.30000000000000004e45

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 40.5%

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

   6. Taylor expanded in x around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. distribute-rgt-neg-in 20.0%

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

      3. *-commutative 20.0%

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

      4. distribute-rgt-neg-in 20.0%

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

      5. distribute-neg-in 20.0%

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

      6. metadata-eval 20.0%

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

      7. distribute-neg-frac 20.0%

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

      8. metadata-eval 20.0%

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

   8. Simplified 20.0%

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

      1. sub-neg 20.0%

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

      2. flip-+ 36.9%

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

      3. metadata-eval 36.9%

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

   10. Applied egg-rr 36.9%

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

   if 1.30000000000000004e45 < x < 1.39999999999999999e117 or 1.40000000000000008e202 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 71.9%

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

   6. Taylor expanded in x around 0 71.9%

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

   7. Taylor expanded in eps around 0 71.9%

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

   if 1.39999999999999999e117 < x < 1.40000000000000008e202

   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 35.2%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. distribute-rgt-neg-in 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-rgt-neg-in 19.9%

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

      5. distribute-neg-in 19.9%

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

      6. metadata-eval 19.9%

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

      7. distribute-neg-frac 19.9%

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

      8. metadata-eval 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around inf 20.1%

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

       1. associate-*r/ 20.1%

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

       2. metadata-eval 20.1%

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

   11. Simplified 20.1%

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

   13. Applied egg-rr 33.7%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+45}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 - \varepsilon \cdot \varepsilon}{1 + \varepsilon} \cdot \left(-1 + \frac{-1}{\varepsilon}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+117}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+117}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+205}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-11)
   (/ (* eps (* (+ x -8.0) (+ -8.0 (* x 2.0)))) 2.0)
   (if (<= x 3.8e-75)
     1.0
     (if (<= x 1.1e+45)
       (/ (/ (- (* eps (+ 2.0 (* x eps))) x) eps) 2.0)
       (if (<= x 4.8e+117)
         0.0
         (if (<= x 1.35e+205)
           (/ (* eps (* (+ x -8.0) (+ x -8.0))) 2.0)
           0.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 3.8e-75) {
		tmp = 1.0;
	} else if (x <= 1.1e+45) {
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0;
	} else if (x <= 4.8e+117) {
		tmp = 0.0;
	} else if (x <= 1.35e+205) {
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-11)) then
        tmp = (eps * ((x + (-8.0d0)) * ((-8.0d0) + (x * 2.0d0)))) / 2.0d0
    else if (x <= 3.8d-75) then
        tmp = 1.0d0
    else if (x <= 1.1d+45) then
        tmp = (((eps * (2.0d0 + (x * eps))) - x) / eps) / 2.0d0
    else if (x <= 4.8d+117) then
        tmp = 0.0d0
    else if (x <= 1.35d+205) then
        tmp = (eps * ((x + (-8.0d0)) * (x + (-8.0d0)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 3.8e-75) {
		tmp = 1.0;
	} else if (x <= 1.1e+45) {
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0;
	} else if (x <= 4.8e+117) {
		tmp = 0.0;
	} else if (x <= 1.35e+205) {
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-11:
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0
	elif x <= 3.8e-75:
		tmp = 1.0
	elif x <= 1.1e+45:
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0
	elif x <= 4.8e+117:
		tmp = 0.0
	elif x <= 1.35e+205:
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(-8.0 + Float64(x * 2.0)))) / 2.0);
	elseif (x <= 3.8e-75)
		tmp = 1.0;
	elseif (x <= 1.1e+45)
		tmp = Float64(Float64(Float64(Float64(eps * Float64(2.0 + Float64(x * eps))) - x) / eps) / 2.0);
	elseif (x <= 4.8e+117)
		tmp = 0.0;
	elseif (x <= 1.35e+205)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(x + -8.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-11)
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	elseif (x <= 3.8e-75)
		tmp = 1.0;
	elseif (x <= 1.1e+45)
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0;
	elseif (x <= 4.8e+117)
		tmp = 0.0;
	elseif (x <= 1.35e+205)
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-11], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(-8.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e-75], 1.0, If[LessEqual[x, 1.1e+45], N[(N[(N[(N[(eps * N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.8e+117], 0.0, If[LessEqual[x, 1.35e+205], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(x + -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.70000000000000005e-11

   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 51.6%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. distribute-rgt-neg-in 23.6%

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

      3. *-commutative 23.6%

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

      4. distribute-rgt-neg-in 23.6%

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

      5. distribute-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

      7. distribute-neg-frac 23.6%

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

      8. metadata-eval 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in eps around inf 23.6%

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

       1. associate-*r/ 23.6%

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

       2. metadata-eval 23.6%

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

   11. Simplified 23.6%

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

   13. Applied egg-rr 39.9%

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

       1. *-commutative 39.9%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + \left(x + -8\right)\right)\right)}}{2} \]

       2. associate-+r+ 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \color{blue}{\left(\left(x + x\right) + -8\right)}\right)}{2} \]

       3. count-2 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(\color{blue}{2 \cdot x} + -8\right)\right)}{2} \]

   15. Simplified 39.9%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(2 \cdot x + -8\right)\right)}}{2} \]

   if -2.70000000000000005e-11 < x < 3.79999999999999994e-75

   1. Initial program 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in x around 0 83.2%

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

   if 3.79999999999999994e-75 < x < 1.1e45

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in x around 0 40.5%

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

   6. Taylor expanded in x around 0 20.0%

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

      1. mul-1-neg 20.0%

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

      2. distribute-rgt-neg-in 20.0%

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

      3. *-commutative 20.0%

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

      4. distribute-rgt-neg-in 20.0%

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

      5. distribute-neg-in 20.0%

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

      6. metadata-eval 20.0%

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

      7. distribute-neg-frac 20.0%

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

      8. metadata-eval 20.0%

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

   8. Simplified 20.0%

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

   9. Taylor expanded in eps around 0 34.5%

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

       1. neg-mul-1 34.5%

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

       2. +-commutative 34.5%

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

       3. unsub-neg 34.5%

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

       4. *-commutative 34.5%

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

   11. Simplified 34.5%

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

   if 1.1e45 < x < 4.7999999999999998e117 or 1.35000000000000006e205 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 71.9%

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

   6. Taylor expanded in x around 0 71.9%

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

   7. Taylor expanded in eps around 0 71.9%

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

   if 4.7999999999999998e117 < x < 1.35000000000000006e205

   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 35.2%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. distribute-rgt-neg-in 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-rgt-neg-in 19.9%

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

      5. distribute-neg-in 19.9%

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

      6. metadata-eval 19.9%

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

      7. distribute-neg-frac 19.9%

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

      8. metadata-eval 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around inf 20.1%

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

       1. associate-*r/ 20.1%

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

       2. metadata-eval 20.1%

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

   11. Simplified 20.1%

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

   13. Applied egg-rr 33.7%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+117}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+205}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+118}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-11)
   (/ (* eps (* (+ x -8.0) (+ -8.0 (* x 2.0)))) 2.0)
   (if (<= x 105000.0)
     1.0
     (if (<= x 3.1e+118)
       0.0
       (if (<= x 4e+202) (/ (* eps (* (+ x -8.0) (+ x -8.0))) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 3.1e+118) {
		tmp = 0.0;
	} else if (x <= 4e+202) {
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-11)) then
        tmp = (eps * ((x + (-8.0d0)) * ((-8.0d0) + (x * 2.0d0)))) / 2.0d0
    else if (x <= 105000.0d0) then
        tmp = 1.0d0
    else if (x <= 3.1d+118) then
        tmp = 0.0d0
    else if (x <= 4d+202) then
        tmp = (eps * ((x + (-8.0d0)) * (x + (-8.0d0)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 3.1e+118) {
		tmp = 0.0;
	} else if (x <= 4e+202) {
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-11:
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0
	elif x <= 105000.0:
		tmp = 1.0
	elif x <= 3.1e+118:
		tmp = 0.0
	elif x <= 4e+202:
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(-8.0 + Float64(x * 2.0)))) / 2.0);
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 3.1e+118)
		tmp = 0.0;
	elseif (x <= 4e+202)
		tmp = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(x + -8.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-11)
		tmp = (eps * ((x + -8.0) * (-8.0 + (x * 2.0)))) / 2.0;
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 3.1e+118)
		tmp = 0.0;
	elseif (x <= 4e+202)
		tmp = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-11], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(-8.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 105000.0], 1.0, If[LessEqual[x, 3.1e+118], 0.0, If[LessEqual[x, 4e+202], N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(x + -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.70000000000000005e-11

   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 51.6%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. distribute-rgt-neg-in 23.6%

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

      3. *-commutative 23.6%

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

      4. distribute-rgt-neg-in 23.6%

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

      5. distribute-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

      7. distribute-neg-frac 23.6%

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

      8. metadata-eval 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in eps around inf 23.6%

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

       1. associate-*r/ 23.6%

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

       2. metadata-eval 23.6%

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

   11. Simplified 23.6%

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

   13. Applied egg-rr 39.9%

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

       1. *-commutative 39.9%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + \left(x + -8\right)\right)\right)}}{2} \]

       2. associate-+r+ 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \color{blue}{\left(\left(x + x\right) + -8\right)}\right)}{2} \]

       3. count-2 39.9%

          \[\leadsto \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(\color{blue}{2 \cdot x} + -8\right)\right)}{2} \]

   15. Simplified 39.9%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(2 \cdot x + -8\right)\right)}}{2} \]

   if -2.70000000000000005e-11 < x < 105000

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around 0 73.2%

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

   if 105000 < x < 3.09999999999999986e118 or 3.9999999999999996e202 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.5%

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

   6. Taylor expanded in x around 0 61.5%

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

   7. Taylor expanded in eps around 0 61.5%

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

   if 3.09999999999999986e118 < x < 3.9999999999999996e202

   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 35.2%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. distribute-rgt-neg-in 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-rgt-neg-in 19.9%

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

      5. distribute-neg-in 19.9%

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

      6. metadata-eval 19.9%

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

      7. distribute-neg-frac 19.9%

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

      8. metadata-eval 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around inf 20.1%

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

       1. associate-*r/ 20.1%

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

       2. metadata-eval 20.1%

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

   11. Simplified 20.1%

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

   13. Applied egg-rr 33.7%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(-8 + x \cdot 2\right)\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+118}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.6% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+117}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (* eps (* (+ x -8.0) (+ x -8.0))) 2.0)))
   (if (<= x -2.7e-11)
     t_0
     (if (<= x 105000.0)
       1.0
       (if (<= x 8e+117) 0.0 (if (<= x 4e+202) t_0 0.0))))))

C

double code(double x, double eps) {
	double t_0 = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	double tmp;
	if (x <= -2.7e-11) {
		tmp = t_0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 8e+117) {
		tmp = 0.0;
	} else if (x <= 4e+202) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (eps * ((x + (-8.0d0)) * (x + (-8.0d0)))) / 2.0d0
    if (x <= (-2.7d-11)) then
        tmp = t_0
    else if (x <= 105000.0d0) then
        tmp = 1.0d0
    else if (x <= 8d+117) then
        tmp = 0.0d0
    else if (x <= 4d+202) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	double tmp;
	if (x <= -2.7e-11) {
		tmp = t_0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 8e+117) {
		tmp = 0.0;
	} else if (x <= 4e+202) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (eps * ((x + -8.0) * (x + -8.0))) / 2.0
	tmp = 0
	if x <= -2.7e-11:
		tmp = t_0
	elif x <= 105000.0:
		tmp = 1.0
	elif x <= 8e+117:
		tmp = 0.0
	elif x <= 4e+202:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * Float64(Float64(x + -8.0) * Float64(x + -8.0))) / 2.0)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = t_0;
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 8e+117)
		tmp = 0.0;
	elseif (x <= 4e+202)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps * ((x + -8.0) * (x + -8.0))) / 2.0;
	tmp = 0.0;
	if (x <= -2.7e-11)
		tmp = t_0;
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 8e+117)
		tmp = 0.0;
	elseif (x <= 4e+202)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * N[(N[(x + -8.0), $MachinePrecision] * N[(x + -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.7e-11], t$95$0, If[LessEqual[x, 105000.0], 1.0, If[LessEqual[x, 8e+117], 0.0, If[LessEqual[x, 4e+202], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.70000000000000005e-11 or 8.0000000000000004e117 < x < 3.9999999999999996e202

   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 45.5%

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

   6. Taylor expanded in x around 0 22.2%

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

      1. mul-1-neg 22.2%

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

      2. distribute-rgt-neg-in 22.2%

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

      3. *-commutative 22.2%

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

      4. distribute-rgt-neg-in 22.2%

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

      5. distribute-neg-in 22.2%

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

      6. metadata-eval 22.2%

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

      7. distribute-neg-frac 22.2%

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

      8. metadata-eval 22.2%

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

   8. Simplified 22.2%

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

   9. Taylor expanded in eps around inf 22.3%

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

       1. associate-*r/ 22.3%

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

       2. metadata-eval 22.3%

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

   11. Simplified 22.3%

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

   13. Applied egg-rr 37.6%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}}{2} \]

   if -2.70000000000000005e-11 < x < 105000

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around 0 73.2%

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

   if 105000 < x < 8.0000000000000004e117 or 3.9999999999999996e202 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.5%

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

   6. Taylor expanded in x around 0 61.5%

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

   7. Taylor expanded in eps around 0 61.5%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+117}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+202}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\left(x + -8\right) \cdot \left(x + -8\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.9% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x \cdot -8 + 64\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+115}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+204}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-11)
   (/ (* eps (+ (* x -8.0) 64.0)) 2.0)
   (if (<= x 105000.0)
     1.0
     (if (<= x 3.5e+115) 0.0 (if (<= x 3e+204) (/ (* x eps) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x * -8.0) + 64.0)) / 2.0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 3.5e+115) {
		tmp = 0.0;
	} else if (x <= 3e+204) {
		tmp = (x * eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-11)) then
        tmp = (eps * ((x * (-8.0d0)) + 64.0d0)) / 2.0d0
    else if (x <= 105000.0d0) then
        tmp = 1.0d0
    else if (x <= 3.5d+115) then
        tmp = 0.0d0
    else if (x <= 3d+204) then
        tmp = (x * eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * ((x * -8.0) + 64.0)) / 2.0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 3.5e+115) {
		tmp = 0.0;
	} else if (x <= 3e+204) {
		tmp = (x * eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-11:
		tmp = (eps * ((x * -8.0) + 64.0)) / 2.0
	elif x <= 105000.0:
		tmp = 1.0
	elif x <= 3.5e+115:
		tmp = 0.0
	elif x <= 3e+204:
		tmp = (x * eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = Float64(Float64(eps * Float64(Float64(x * -8.0) + 64.0)) / 2.0);
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 3.5e+115)
		tmp = 0.0;
	elseif (x <= 3e+204)
		tmp = Float64(Float64(x * eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-11)
		tmp = (eps * ((x * -8.0) + 64.0)) / 2.0;
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 3.5e+115)
		tmp = 0.0;
	elseif (x <= 3e+204)
		tmp = (x * eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-11], N[(N[(eps * N[(N[(x * -8.0), $MachinePrecision] + 64.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 105000.0], 1.0, If[LessEqual[x, 3.5e+115], 0.0, If[LessEqual[x, 3e+204], N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.70000000000000005e-11

   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 51.6%

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

   6. Taylor expanded in x around 0 23.6%

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

      1. mul-1-neg 23.6%

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

      2. distribute-rgt-neg-in 23.6%

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

      3. *-commutative 23.6%

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

      4. distribute-rgt-neg-in 23.6%

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

      5. distribute-neg-in 23.6%

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

      6. metadata-eval 23.6%

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

      7. distribute-neg-frac 23.6%

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

      8. metadata-eval 23.6%

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

   8. Simplified 23.6%

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

   9. Taylor expanded in eps around inf 23.6%

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

       1. associate-*r/ 23.6%

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

       2. metadata-eval 23.6%

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

   11. Simplified 23.6%

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

   13. Applied egg-rr 31.7%

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

       1. distribute-lft-in 31.7%

          \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(-8 \cdot x + -8 \cdot -8\right)}}{2} \]

       2. metadata-eval 31.7%

          \[\leadsto \frac{\varepsilon \cdot \left(-8 \cdot x + \color{blue}{64}\right)}{2} \]

   15. Simplified 31.7%

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\left(-8 \cdot x + 64\right)}}{2} \]

   if -2.70000000000000005e-11 < x < 105000

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around 0 73.2%

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

   if 105000 < x < 3.50000000000000005e115 or 2.99999999999999983e204 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.5%

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

   6. Taylor expanded in x around 0 61.5%

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

   7. Taylor expanded in eps around 0 61.5%

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

   if 3.50000000000000005e115 < x < 2.99999999999999983e204

   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 35.2%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. distribute-rgt-neg-in 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-rgt-neg-in 19.9%

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

      5. distribute-neg-in 19.9%

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

      6. metadata-eval 19.9%

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

      7. distribute-neg-frac 19.9%

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

      8. metadata-eval 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around inf 20.1%

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

       1. associate-*r/ 20.1%

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

       2. metadata-eval 20.1%

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

   11. Simplified 20.1%

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

   12. Taylor expanded in eps around inf 20.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x \cdot -8 + 64\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+115}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+204}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+120}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+203}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-11)
   (/ (* eps (- x)) 2.0)
   (if (<= x 105000.0)
     1.0
     (if (<= x 2.1e+120) 0.0 (if (<= x 3e+203) (/ (* x eps) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 2.1e+120) {
		tmp = 0.0;
	} else if (x <= 3e+203) {
		tmp = (x * eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-11)) then
        tmp = (eps * -x) / 2.0d0
    else if (x <= 105000.0d0) then
        tmp = 1.0d0
    else if (x <= 2.1d+120) then
        tmp = 0.0d0
    else if (x <= 3d+203) then
        tmp = (x * eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-11) {
		tmp = (eps * -x) / 2.0;
	} else if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 2.1e+120) {
		tmp = 0.0;
	} else if (x <= 3e+203) {
		tmp = (x * eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-11:
		tmp = (eps * -x) / 2.0
	elif x <= 105000.0:
		tmp = 1.0
	elif x <= 2.1e+120:
		tmp = 0.0
	elif x <= 3e+203:
		tmp = (x * eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-11)
		tmp = Float64(Float64(eps * Float64(-x)) / 2.0);
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 2.1e+120)
		tmp = 0.0;
	elseif (x <= 3e+203)
		tmp = Float64(Float64(x * eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-11)
		tmp = (eps * -x) / 2.0;
	elseif (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 2.1e+120)
		tmp = 0.0;
	elseif (x <= 3e+203)
		tmp = (x * eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-11], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 105000.0], 1.0, If[LessEqual[x, 2.1e+120], 0.0, If[LessEqual[x, 3e+203], N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.70000000000000005e-11

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 59.9%

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

   6. Taylor expanded in eps around inf 31.7%

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

      1. associate-*r* 31.7%

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

      2. mul-1-neg 31.7%

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

   8. Simplified 31.7%

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

   if -2.70000000000000005e-11 < x < 105000

   1. Initial program 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around 0 73.2%

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

   if 105000 < x < 2.1e120 or 3e203 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.5%

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

   6. Taylor expanded in x around 0 61.5%

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

   7. Taylor expanded in eps around 0 61.5%

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

   if 2.1e120 < x < 3e203

   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 35.2%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. distribute-rgt-neg-in 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-rgt-neg-in 19.9%

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

      5. distribute-neg-in 19.9%

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

      6. metadata-eval 19.9%

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

      7. distribute-neg-frac 19.9%

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

      8. metadata-eval 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around inf 20.1%

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

       1. associate-*r/ 20.1%

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

       2. metadata-eval 20.1%

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

   11. Simplified 20.1%

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

   12. Taylor expanded in eps around inf 20.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{elif}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+120}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+203}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+120}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+204}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 105000.0)
   1.0
   (if (<= x 7e+120) 0.0 (if (<= x 1.75e+204) (/ (* x eps) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 7e+120) {
		tmp = 0.0;
	} else if (x <= 1.75e+204) {
		tmp = (x * eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 105000.0d0) then
        tmp = 1.0d0
    else if (x <= 7d+120) then
        tmp = 0.0d0
    else if (x <= 1.75d+204) then
        tmp = (x * eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 7e+120) {
		tmp = 0.0;
	} else if (x <= 1.75e+204) {
		tmp = (x * eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 105000.0:
		tmp = 1.0
	elif x <= 7e+120:
		tmp = 0.0
	elif x <= 1.75e+204:
		tmp = (x * eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 7e+120)
		tmp = 0.0;
	elseif (x <= 1.75e+204)
		tmp = Float64(Float64(x * eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 7e+120)
		tmp = 0.0;
	elseif (x <= 1.75e+204)
		tmp = (x * eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 105000.0], 1.0, If[LessEqual[x, 7e+120], 0.0, If[LessEqual[x, 1.75e+204], N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 105000

   1. Initial program 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around 0 55.6%

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

   if 105000 < x < 7.00000000000000015e120 or 1.74999999999999995e204 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.5%

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

   6. Taylor expanded in x around 0 61.5%

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

   7. Taylor expanded in eps around 0 61.5%

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

   if 7.00000000000000015e120 < x < 1.74999999999999995e204

   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 35.2%

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

   6. Taylor expanded in x around 0 19.9%

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

      1. mul-1-neg 19.9%

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

      2. distribute-rgt-neg-in 19.9%

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

      3. *-commutative 19.9%

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

      4. distribute-rgt-neg-in 19.9%

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

      5. distribute-neg-in 19.9%

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

      6. metadata-eval 19.9%

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

      7. distribute-neg-frac 19.9%

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

      8. metadata-eval 19.9%

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

   8. Simplified 19.9%

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

   9. Taylor expanded in eps around inf 20.1%

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

       1. associate-*r/ 20.1%

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

       2. metadata-eval 20.1%

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

   11. Simplified 20.1%

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

   12. Taylor expanded in eps around inf 20.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+120}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+204}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 105000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 105000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 105000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 105000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 105000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 105000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 105000

   1. Initial program 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around 0 55.6%

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

   if 105000 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.4%

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

   6. Taylor expanded in x around 0 47.4%

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

   7. Taylor expanded in eps around 0 47.4%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 16.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 77.7%

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

3. Simplified 65.0%

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

5. Taylor expanded in eps around 0 15.0%

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

6. Taylor expanded in x around 0 15.2%

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

7. Taylor expanded in eps around 0 15.2%

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

8. Final simplification 15.2%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))