NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 98.9%

Time: 22.7s

Alternatives: 16

Speedup: 1.1×

• 38.9% 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: 73.3% 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: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (exp((x * (-1.0 + eps))) + 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 * ((-1.0d0) + eps))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.3%

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

3. Simplified 66.2%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Taylor expanded in eps around -inf 99.1%

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

   1. associate-*r* 99.1%

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

   2. *-commutative 99.1%

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

   3. associate-*l* 99.1%

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

   4. cancel-sign-sub-inv 99.1%

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

   5. metadata-eval 99.1%

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

   6. *-lft-identity 99.1%

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

   7. distribute-lft-in 99.1%

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

   8. metadata-eval 99.1%

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

   9. mul-1-neg 99.1%

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

   10. unsub-neg 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 2: 73.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.55 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(t\_0 + x \cdot \left(t\_0 \cdot \left(-1 + \varepsilon\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(2 \cdot \frac{1}{\varepsilon} + x \cdot \left(t\_0 \cdot \left(-1 - \varepsilon\right) + \left(t\_0 \cdot \left(\varepsilon + 1\right) + x \cdot \left(t\_0 \cdot {\left(\varepsilon + 1\right)}^{2}\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<= eps 1.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
     (if (<= eps 3.55e+103)
       (/
        (+
         (+ t_0 (* x (* t_0 (+ -1.0 eps))))
         (* (exp (* x (- -1.0 eps))) (- (/ -1.0 eps) -1.0)))
        2.0)
       (/
        (+
         2.0
         (+
          (* 2.0 (/ 1.0 eps))
          (*
           x
           (+
            (* t_0 (- -1.0 eps))
            (+ (* t_0 (+ eps 1.0)) (* x (* t_0 (pow (+ eps 1.0) 2.0))))))))
        2.0)))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (eps <= 1.0) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 3.55e+103) {
		tmp = ((t_0 + (x * (t_0 * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	} else {
		tmp = (2.0 + ((2.0 * (1.0 / eps)) + (x * ((t_0 * (-1.0 - eps)) + ((t_0 * (eps + 1.0)) + (x * (t_0 * pow((eps + 1.0), 2.0)))))))) / 2.0;
	}
	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 = 1.0d0 + (1.0d0 / eps)
    if (eps <= 1.0d0) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else if (eps <= 3.55d+103) then
        tmp = ((t_0 + (x * (t_0 * ((-1.0d0) + eps)))) + (exp((x * ((-1.0d0) - eps))) * (((-1.0d0) / eps) - (-1.0d0)))) / 2.0d0
    else
        tmp = (2.0d0 + ((2.0d0 * (1.0d0 / eps)) + (x * ((t_0 * ((-1.0d0) - eps)) + ((t_0 * (eps + 1.0d0)) + (x * (t_0 * ((eps + 1.0d0) ** 2.0d0)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (eps <= 1.0) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else if (eps <= 3.55e+103) {
		tmp = ((t_0 + (x * (t_0 * (-1.0 + eps)))) + (Math.exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	} else {
		tmp = (2.0 + ((2.0 * (1.0 / eps)) + (x * ((t_0 * (-1.0 - eps)) + ((t_0 * (eps + 1.0)) + (x * (t_0 * Math.pow((eps + 1.0), 2.0)))))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + (1.0 / eps)
	tmp = 0
	if eps <= 1.0:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	elif eps <= 3.55e+103:
		tmp = ((t_0 + (x * (t_0 * (-1.0 + eps)))) + (math.exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0
	else:
		tmp = (2.0 + ((2.0 * (1.0 / eps)) + (x * ((t_0 * (-1.0 - eps)) + ((t_0 * (eps + 1.0)) + (x * (t_0 * math.pow((eps + 1.0), 2.0)))))))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	elseif (eps <= 3.55e+103)
		tmp = Float64(Float64(Float64(t_0 + Float64(x * Float64(t_0 * Float64(-1.0 + eps)))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(Float64(-1.0 / eps) - -1.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(2.0 * Float64(1.0 / eps)) + Float64(x * Float64(Float64(t_0 * Float64(-1.0 - eps)) + Float64(Float64(t_0 * Float64(eps + 1.0)) + Float64(x * Float64(t_0 * (Float64(eps + 1.0) ^ 2.0)))))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 + (1.0 / eps);
	tmp = 0.0;
	if (eps <= 1.0)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	elseif (eps <= 3.55e+103)
		tmp = ((t_0 + (x * (t_0 * (-1.0 + eps)))) + (exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	else
		tmp = (2.0 + ((2.0 * (1.0 / eps)) + (x * ((t_0 * (-1.0 - eps)) + ((t_0 * (eps + 1.0)) + (x * (t_0 * ((eps + 1.0) ^ 2.0)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 3.55e+103], N[(N[(N[(t$95$0 + N[(x * N[(t$95$0 * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(N[(2.0 * N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(t$95$0 * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] + N[(x * N[(t$95$0 * N[Power[N[(eps + 1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 67.2%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in eps around 0 36.3%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in eps around 0 70.2%

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

   if 1 < eps < 3.5500000000000001e103

   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 73.8%

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

      1. +-commutative 73.8%

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

      2. associate-+r+ 73.8%

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

      3. mul-1-neg 73.8%

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

      4. distribute-rgt-neg-in 73.8%

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

      5. *-commutative 73.8%

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

      6. distribute-rgt-neg-in 73.8%

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

      7. distribute-neg-in 73.8%

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

      8. metadata-eval 73.8%

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

      9. distribute-neg-frac 73.8%

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

      10. metadata-eval 73.8%

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

   7. Simplified 73.8%

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

   if 3.5500000000000001e103 < eps

   1. Initial program 100.0%

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

   3. Simplified 86.0%

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

      1. add-cube-cbrt 85.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}}}{2} \]

      2. pow3 85.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right)}^{3}}}{2} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]

   7. Taylor expanded in x around 0 89.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 3.55 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 + \varepsilon\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(2 \cdot \frac{1}{\varepsilon} + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(-1 - \varepsilon\right) + \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon + 1\right) + x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon + 1\right)}^{2}\right)\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-13} \lor \neg \left(x \leq 68000000000000 \lor \neg \left(x \leq 4.2 \cdot 10^{+67}\right) \land x \leq 1.15 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-261)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (or (<= x 1.65e-13)
           (not
            (or (<= x 68000000000000.0)
                (and (not (<= x 4.2e+67)) (<= x 1.15e+105)))))
     (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-261) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 1.65e-13) || !((x <= 68000000000000.0) || (!(x <= 4.2e+67) && (x <= 1.15e+105)))) {
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	} else {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1d-261)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 1.65d-13) .or. (.not. (x <= 68000000000000.0d0) .or. (.not. (x <= 4.2d+67)) .and. (x <= 1.15d+105))) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    else
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-261) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 1.65e-13) || !((x <= 68000000000000.0) || (!(x <= 4.2e+67) && (x <= 1.15e+105)))) {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	} else {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1e-261:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 1.65e-13) or not ((x <= 68000000000000.0) or (not (x <= 4.2e+67) and (x <= 1.15e+105))):
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	else:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-261)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 1.65e-13) || !((x <= 68000000000000.0) || (!(x <= 4.2e+67) && (x <= 1.15e+105))))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-261)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 1.65e-13) || ~(((x <= 68000000000000.0) || (~((x <= 4.2e+67)) && (x <= 1.15e+105)))))
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	else
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-261], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.65e-13], N[Not[Or[LessEqual[x, 68000000000000.0], And[N[Not[LessEqual[x, 4.2e+67]], $MachinePrecision], LessEqual[x, 1.15e+105]]]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999984e-262

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 49.8%

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

   6. Taylor expanded in eps around inf 75.8%

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

      1. cancel-sign-sub-inv 75.8%

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

      2. metadata-eval 75.8%

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

      3. *-lft-identity 75.8%

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

      4. associate-*r* 75.8%

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

      5. +-commutative 75.8%

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

      6. exp-prod 66.1%

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

      7. +-commutative 66.1%

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

      8. *-lft-identity 66.1%

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

      9. metadata-eval 66.1%

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

      10. cancel-sign-sub-inv 66.1%

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

      11. exp-prod 75.8%

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

      12. associate-*r* 75.8%

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

      13. mul-1-neg 75.8%

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

      14. mul-1-neg 75.8%

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

      15. associate-*r* 75.8%

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

      16. *-commutative 75.8%

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

      17. associate-*l* 75.8%

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

      18. cancel-sign-sub-inv 75.8%

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

      19. metadata-eval 75.8%

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

      20. *-lft-identity 75.8%

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

      21. distribute-lft-in 75.8%

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

   8. Simplified 75.8%

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

   if -9.99999999999999984e-262 < x < 1.65e-13 or 6.8e13 < x < 4.2000000000000003e67 or 1.1499999999999999e105 < x

   1. Initial program 79.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in eps around 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 -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. mul-1-neg 100.0%

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

      10. 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%

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

   9. Taylor expanded in x around 0 60.3%

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

   if 1.65e-13 < x < 6.8e13 or 4.2000000000000003e67 < x < 1.1499999999999999e105

   1. Initial program 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in eps around 0 73.4%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in eps around 0 90.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-261}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-13} \lor \neg \left(x \leq 68000000000000 \lor \neg \left(x \leq 4.2 \cdot 10^{+67}\right) \land x \leq 1.15 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 + \varepsilon\right)}\\ t_1 := x \cdot \left(-1 - \varepsilon\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + e^{t\_1}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_0 + \left(1 + t\_1\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+14} \lor \neg \left(x \leq 1.85 \cdot 10^{+67}\right) \land x \leq 2.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (+ -1.0 eps)))) (t_1 (* x (- -1.0 eps))))
   (if (<= x -1e-262)
     (/ (+ 1.0 (exp t_1)) 2.0)
     (if (<= x 1.65e-13)
       (/ (+ t_0 (+ 1.0 t_1)) 2.0)
       (if (or (<= x 1.8e+14) (and (not (<= x 1.85e+67)) (<= x 2.7e+103)))
         (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
         (/ (+ 1.0 t_0) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 + eps)));
	double t_1 = x * (-1.0 - eps);
	double tmp;
	if (x <= -1e-262) {
		tmp = (1.0 + exp(t_1)) / 2.0;
	} else if (x <= 1.65e-13) {
		tmp = (t_0 + (1.0 + t_1)) / 2.0;
	} else if ((x <= 1.8e+14) || (!(x <= 1.85e+67) && (x <= 2.7e+103))) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) + eps)))
    t_1 = x * ((-1.0d0) - eps)
    if (x <= (-1d-262)) then
        tmp = (1.0d0 + exp(t_1)) / 2.0d0
    else if (x <= 1.65d-13) then
        tmp = (t_0 + (1.0d0 + t_1)) / 2.0d0
    else if ((x <= 1.8d+14) .or. (.not. (x <= 1.85d+67)) .and. (x <= 2.7d+103)) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * (-1.0 + eps)));
	double t_1 = x * (-1.0 - eps);
	double tmp;
	if (x <= -1e-262) {
		tmp = (1.0 + Math.exp(t_1)) / 2.0;
	} else if (x <= 1.65e-13) {
		tmp = (t_0 + (1.0 + t_1)) / 2.0;
	} else if ((x <= 1.8e+14) || (!(x <= 1.85e+67) && (x <= 2.7e+103))) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * (-1.0 + eps)))
	t_1 = x * (-1.0 - eps)
	tmp = 0
	if x <= -1e-262:
		tmp = (1.0 + math.exp(t_1)) / 2.0
	elif x <= 1.65e-13:
		tmp = (t_0 + (1.0 + t_1)) / 2.0
	elif (x <= 1.8e+14) or (not (x <= 1.85e+67) and (x <= 2.7e+103)):
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(-1.0 + eps)))
	t_1 = Float64(x * Float64(-1.0 - eps))
	tmp = 0.0
	if (x <= -1e-262)
		tmp = Float64(Float64(1.0 + exp(t_1)) / 2.0);
	elseif (x <= 1.65e-13)
		tmp = Float64(Float64(t_0 + Float64(1.0 + t_1)) / 2.0);
	elseif ((x <= 1.8e+14) || (!(x <= 1.85e+67) && (x <= 2.7e+103)))
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * (-1.0 + eps)));
	t_1 = x * (-1.0 - eps);
	tmp = 0.0;
	if (x <= -1e-262)
		tmp = (1.0 + exp(t_1)) / 2.0;
	elseif (x <= 1.65e-13)
		tmp = (t_0 + (1.0 + t_1)) / 2.0;
	elseif ((x <= 1.8e+14) || (~((x <= 1.85e+67)) && (x <= 2.7e+103)))
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-262], N[(N[(1.0 + N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-13], N[(N[(t$95$0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.8e+14], And[N[Not[LessEqual[x, 1.85e+67]], $MachinePrecision], LessEqual[x, 2.7e+103]]], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000001e-262

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 49.8%

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

   6. Taylor expanded in eps around inf 75.8%

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

      1. cancel-sign-sub-inv 75.8%

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

      2. metadata-eval 75.8%

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

      3. *-lft-identity 75.8%

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

      4. associate-*r* 75.8%

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

      5. +-commutative 75.8%

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

      6. exp-prod 66.1%

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

      7. +-commutative 66.1%

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

      8. *-lft-identity 66.1%

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

      9. metadata-eval 66.1%

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

      10. cancel-sign-sub-inv 66.1%

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

      11. exp-prod 75.8%

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

      12. associate-*r* 75.8%

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

      13. mul-1-neg 75.8%

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

      14. mul-1-neg 75.8%

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

      15. associate-*r* 75.8%

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

      16. *-commutative 75.8%

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

      17. associate-*l* 75.8%

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

      18. cancel-sign-sub-inv 75.8%

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

      19. metadata-eval 75.8%

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

      20. *-lft-identity 75.8%

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

      21. distribute-lft-in 75.8%

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

   8. Simplified 75.8%

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

   if -1.00000000000000001e-262 < x < 1.65e-13

   1. Initial program 56.4%

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

   3. Simplified 35.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{x \cdot \left(\varepsilon + -1\right)}, {\left(e^{1 + \varepsilon}\right)}^{\left(-x\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\right)}{2}} \]
   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. Taylor expanded in x around 0 87.0%

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

   if 1.65e-13 < x < 1.8e14 or 1.8499999999999999e67 < x < 2.69999999999999993e103

   1. Initial program 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in eps around 0 73.4%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in eps around 0 90.9%

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

   if 1.8e14 < x < 1.8499999999999999e67 or 2.69999999999999993e103 < 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 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 -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. mul-1-neg 100.0%

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

      10. 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%

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

   9. Taylor expanded in x around 0 37.3%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-262}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+14} \lor \neg \left(x \leq 1.85 \cdot 10^{+67}\right) \land x \leq 2.7 \cdot 10^{+103}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+66} \lor \neg \left(x \leq 1.2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.6e-263)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (or (<= x 3.5e+66) (not (<= x 1.2e+107)))
     (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.6e-263) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 3.5e+66) || !(x <= 1.2e+107)) {
		tmp = (1.0 + exp((x * (-1.0 + 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 <= (-9.6d-263)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 3.5d+66) .or. (.not. (x <= 1.2d+107))) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) + 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 <= -9.6e-263) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 3.5e+66) || !(x <= 1.2e+107)) {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9.6e-263:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 3.5e+66) or not (x <= 1.2e+107):
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.6e-263)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 3.5e+66) || !(x <= 1.2e+107))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9.6e-263)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 3.5e+66) || ~((x <= 1.2e+107)))
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.6e-263], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.5e+66], N[Not[LessEqual[x, 1.2e+107]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -9.6000000000000001e-263

   1. Initial program 73.5%

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

   3. Simplified 73.5%

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

   5. Taylor expanded in x around 0 49.8%

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

   6. Taylor expanded in eps around inf 75.8%

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

      1. cancel-sign-sub-inv 75.8%

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

      2. metadata-eval 75.8%

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

      3. *-lft-identity 75.8%

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

      4. associate-*r* 75.8%

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

      5. +-commutative 75.8%

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

      6. exp-prod 66.1%

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

      7. +-commutative 66.1%

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

      8. *-lft-identity 66.1%

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

      9. metadata-eval 66.1%

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

      10. cancel-sign-sub-inv 66.1%

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

      11. exp-prod 75.8%

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

      12. associate-*r* 75.8%

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

      13. mul-1-neg 75.8%

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

      14. mul-1-neg 75.8%

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

      15. associate-*r* 75.8%

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

      16. *-commutative 75.8%

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

      17. associate-*l* 75.8%

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

      18. cancel-sign-sub-inv 75.8%

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

      19. metadata-eval 75.8%

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

      20. *-lft-identity 75.8%

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

      21. distribute-lft-in 75.8%

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

   8. Simplified 75.8%

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

   if -9.6000000000000001e-263 < x < 3.4999999999999997e66 or 1.2e107 < x

   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 68.3%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around -inf 98.7%

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

      1. associate-*r* 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*l* 98.7%

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

      4. cancel-sign-sub-inv 98.7%

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

      5. metadata-eval 98.7%

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

      6. *-lft-identity 98.7%

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

      7. distribute-lft-in 98.7%

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

      8. metadata-eval 98.7%

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

      9. mul-1-neg 98.7%

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

      10. unsub-neg 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around 0 59.0%

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

   if 3.4999999999999997e66 < x < 1.2e107

   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 86.9%

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

      1. mul-1-neg 86.9%

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

      2. mul-1-neg 86.9%

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

      3. rec-exp 86.9%

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

      4. sub-neg 86.9%

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

      5. div-sub 86.9%

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

      6. mul-1-neg 86.9%

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

      7. rec-exp 86.9%

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

      8. +-inverses 86.9%

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

   7. Simplified 86.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-263}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+66} \lor \neg \left(x \leq 1.2 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+53} \lor \neg \left(x \leq 5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{\frac{e^{x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 68000000000000.0)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0)
   (if (or (<= x 1.4e+53) (not (<= x 5e+103))) (/ (/ (exp x) eps) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 68000000000000.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if ((x <= 1.4e+53) || !(x <= 5e+103)) {
		tmp = (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 <= 68000000000000.0d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    else if ((x <= 1.4d+53) .or. (.not. (x <= 5d+103))) then
        tmp = (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 <= 68000000000000.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if ((x <= 1.4e+53) || !(x <= 5e+103)) {
		tmp = (Math.exp(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 68000000000000.0:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	elif (x <= 1.4e+53) or not (x <= 5e+103):
		tmp = (math.exp(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 68000000000000.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	elseif ((x <= 1.4e+53) || !(x <= 5e+103))
		tmp = Float64(Float64(exp(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 68000000000000.0)
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	elseif ((x <= 1.4e+53) || ~((x <= 5e+103)))
		tmp = (exp(x) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 68000000000000.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.4e+53], N[Not[LessEqual[x, 5e+103]], $MachinePrecision]], N[(N[(N[Exp[x], $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 6.8e13

   1. Initial program 66.4%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around 0 55.4%

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

   6. Taylor expanded in eps around 0 59.1%

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

   if 6.8e13 < x < 1.4e53 or 5e103 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}}}{2} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right)}^{3}}}{2} \]

   6. Applied egg-rr 64.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]

   7. Taylor expanded in eps around 0 36.9%

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

   if 1.4e53 < x < 5e103

   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 82.6%

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

      1. mul-1-neg 82.6%

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

      2. mul-1-neg 82.6%

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

      3. rec-exp 82.6%

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

      4. sub-neg 82.6%

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

      5. div-sub 82.6%

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

      6. mul-1-neg 82.6%

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

      7. rec-exp 82.6%

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

      8. +-inverses 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+53} \lor \neg \left(x \leq 5 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{\frac{e^{x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+52} \lor \neg \left(x \leq 10^{+104}\right):\\ \;\;\;\;\frac{\frac{e^{x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 68000000000000.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 1.2e+52) (not (<= x 1e+104))) (/ (/ (exp x) eps) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 68000000000000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 1.2e+52) || !(x <= 1e+104)) {
		tmp = (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 <= 68000000000000.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 1.2d+52) .or. (.not. (x <= 1d+104))) then
        tmp = (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 <= 68000000000000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 1.2e+52) || !(x <= 1e+104)) {
		tmp = (Math.exp(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 68000000000000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 1.2e+52) or not (x <= 1e+104):
		tmp = (math.exp(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 68000000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 1.2e+52) || !(x <= 1e+104))
		tmp = Float64(Float64(exp(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 68000000000000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 1.2e+52) || ~((x <= 1e+104)))
		tmp = (exp(x) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 68000000000000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.2e+52], N[Not[LessEqual[x, 1e+104]], $MachinePrecision]], N[(N[(N[Exp[x], $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 6.8e13

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 44.8%

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

   6. Taylor expanded in eps around inf 77.2%

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

      1. cancel-sign-sub-inv 77.2%

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

      2. metadata-eval 77.2%

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

      3. *-lft-identity 77.2%

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

      4. associate-*r* 77.2%

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

      5. +-commutative 77.2%

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

      6. exp-prod 67.6%

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

      7. +-commutative 67.6%

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

      8. *-lft-identity 67.6%

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

      9. metadata-eval 67.6%

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

      10. cancel-sign-sub-inv 67.6%

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

      11. exp-prod 77.2%

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

      12. associate-*r* 77.2%

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

      13. mul-1-neg 77.2%

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

      14. mul-1-neg 77.2%

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

      15. associate-*r* 77.2%

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

      16. *-commutative 77.2%

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

      17. associate-*l* 77.2%

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

      18. cancel-sign-sub-inv 77.2%

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

      19. metadata-eval 77.2%

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

      20. *-lft-identity 77.2%

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

      21. distribute-lft-in 77.2%

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

   8. Simplified 77.2%

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

   9. Taylor expanded in eps around 0 73.7%

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

   if 6.8e13 < x < 1.2e52 or 1e104 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}}}{2} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right)}^{3}}}{2} \]

   6. Applied egg-rr 64.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]

   7. Taylor expanded in eps around 0 36.9%

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

   if 1.2e52 < x < 1e104

   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 82.6%

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

      1. mul-1-neg 82.6%

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

      2. mul-1-neg 82.6%

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

      3. rec-exp 82.6%

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

      4. sub-neg 82.6%

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

      5. div-sub 82.6%

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

      6. mul-1-neg 82.6%

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

      7. rec-exp 82.6%

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

      8. +-inverses 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+52} \lor \neg \left(x \leq 10^{+104}\right):\\ \;\;\;\;\frac{\frac{e^{x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+52} \lor \neg \left(x \leq 1.2 \cdot 10^{+106}\right):\\ \;\;\;\;\frac{\frac{e^{x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 68000000000000.0)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (or (<= x 4.6e+52) (not (<= x 1.2e+106))) (/ (/ (exp x) eps) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 68000000000000.0) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 4.6e+52) || !(x <= 1.2e+106)) {
		tmp = (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 <= 68000000000000.0d0) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 4.6d+52) .or. (.not. (x <= 1.2d+106))) then
        tmp = (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 <= 68000000000000.0) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 4.6e+52) || !(x <= 1.2e+106)) {
		tmp = (Math.exp(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 68000000000000.0:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 4.6e+52) or not (x <= 1.2e+106):
		tmp = (math.exp(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 68000000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 4.6e+52) || !(x <= 1.2e+106))
		tmp = Float64(Float64(exp(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 68000000000000.0)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 4.6e+52) || ~((x <= 1.2e+106)))
		tmp = (exp(x) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 68000000000000.0], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4.6e+52], N[Not[LessEqual[x, 1.2e+106]], $MachinePrecision]], N[(N[(N[Exp[x], $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 6.8e13

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 44.8%

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

   6. Taylor expanded in eps around inf 77.2%

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

      1. cancel-sign-sub-inv 77.2%

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

      2. metadata-eval 77.2%

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

      3. *-lft-identity 77.2%

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

      4. associate-*r* 77.2%

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

      5. +-commutative 77.2%

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

      6. exp-prod 67.6%

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

      7. +-commutative 67.6%

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

      8. *-lft-identity 67.6%

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

      9. metadata-eval 67.6%

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

      10. cancel-sign-sub-inv 67.6%

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

      11. exp-prod 77.2%

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

      12. associate-*r* 77.2%

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

      13. mul-1-neg 77.2%

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

      14. mul-1-neg 77.2%

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

      15. associate-*r* 77.2%

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

      16. *-commutative 77.2%

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

      17. associate-*l* 77.2%

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

      18. cancel-sign-sub-inv 77.2%

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

      19. metadata-eval 77.2%

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

      20. *-lft-identity 77.2%

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

      21. distribute-lft-in 77.2%

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

   8. Simplified 77.2%

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

   if 6.8e13 < x < 4.6e52 or 1.2e106 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}}}{2} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 + \frac{-1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\right)}^{3}}}{2} \]

   6. Applied egg-rr 64.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right)}\right)}^{3}}}{2} \]

   7. Taylor expanded in eps around 0 36.9%

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

   if 4.6e52 < x < 1.2e106

   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 82.6%

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

      1. mul-1-neg 82.6%

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

      2. mul-1-neg 82.6%

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

      3. rec-exp 82.6%

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

      4. sub-neg 82.6%

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

      5. div-sub 82.6%

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

      6. mul-1-neg 82.6%

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

      7. rec-exp 82.6%

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

      8. +-inverses 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 68000000000000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+52} \lor \neg \left(x \leq 1.2 \cdot 10^{+106}\right):\\ \;\;\;\;\frac{\frac{e^{x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.85)
   (* (* x eps) -0.5)
   (if (<= x 500.0) 1.0 (if (<= x 7.5e+217) 0.0 (/ (* x eps) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.85) {
		tmp = (x * eps) * -0.5;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.85d0)) then
        tmp = (x * eps) * (-0.5d0)
    else if (x <= 500.0d0) then
        tmp = 1.0d0
    else if (x <= 7.5d+217) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.85) {
		tmp = (x * eps) * -0.5;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.85:
		tmp = (x * eps) * -0.5
	elif x <= 500.0:
		tmp = 1.0
	elif x <= 7.5e+217:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.85)
		tmp = Float64(Float64(x * eps) * -0.5);
	elseif (x <= 500.0)
		tmp = 1.0;
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.85)
		tmp = (x * eps) * -0.5;
	elseif (x <= 500.0)
		tmp = 1.0;
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.85], N[(N[(x * eps), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 500.0], 1.0, If[LessEqual[x, 7.5e+217], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.849999999999999978

   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 66.5%

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

   7. Taylor expanded in eps around inf 32.3%

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

      1. frac-2neg 32.3%

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

      2. div-inv 32.3%

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

      3. distribute-rgt-neg-in 32.3%

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

      4. add-sqr-sqrt 32.3%

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

      5. sqrt-unprod 32.3%

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

      6. sqr-neg 32.3%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 24.7%

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

      9. *-commutative 24.7%

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

      10. metadata-eval 24.7%

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

      11. metadata-eval 24.7%

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

   9. Applied egg-rr 24.7%

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

   if -0.849999999999999978 < x < 500

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in x around 0 70.1%

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

   if 500 < x < 7.5000000000000001e217

   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 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

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

      3. rec-exp 55.1%

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

      4. sub-neg 55.1%

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

      5. div-sub 55.1%

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

      6. mul-1-neg 55.1%

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

      7. rec-exp 55.1%

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

      8. +-inverses 55.1%

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

   7. Simplified 55.1%

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

   if 7.5000000000000001e217 < 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 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 39.1%

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

   7. Taylor expanded in eps around inf 35.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.5)
   (/ (* x (- -1.0 eps)) 2.0)
   (if (<= x 500.0) 1.0 (if (<= x 7.5e+217) 0.0 (/ (* x eps) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.5) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.5d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else if (x <= 500.0d0) then
        tmp = 1.0d0
    else if (x <= 7.5d+217) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.5) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 500.0) {
		tmp = 1.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.5:
		tmp = (x * (-1.0 - eps)) / 2.0
	elif x <= 500.0:
		tmp = 1.0
	elif x <= 7.5e+217:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.5)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	elseif (x <= 500.0)
		tmp = 1.0;
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.5)
		tmp = (x * (-1.0 - eps)) / 2.0;
	elseif (x <= 500.0)
		tmp = 1.0;
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.5], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 500.0], 1.0, If[LessEqual[x, 7.5e+217], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.5

   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 58.5%

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

   6. Taylor expanded in eps around inf 58.5%

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

      1. cancel-sign-sub-inv 58.5%

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

      2. metadata-eval 58.5%

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

      3. *-lft-identity 58.5%

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

      4. associate-*r* 58.5%

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

      5. +-commutative 58.5%

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

      6. exp-prod 58.5%

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

      7. +-commutative 58.5%

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

      8. *-lft-identity 58.5%

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

      9. metadata-eval 58.5%

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

      10. cancel-sign-sub-inv 58.5%

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

      11. exp-prod 58.5%

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

      12. associate-*r* 58.5%

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

      13. mul-1-neg 58.5%

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

      14. mul-1-neg 58.5%

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

      15. associate-*r* 58.5%

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

      16. *-commutative 58.5%

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

      17. associate-*l* 58.5%

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

      18. cancel-sign-sub-inv 58.5%

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

      19. metadata-eval 58.5%

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

      20. *-lft-identity 58.5%

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

      21. distribute-lft-in 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in x around 0 24.7%

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

   10. Taylor expanded in x around inf 24.7%

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

       1. *-commutative 24.7%

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

       2. associate-*r* 24.7%

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

       3. distribute-lft-in 24.7%

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

       4. metadata-eval 24.7%

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

       5. neg-mul-1 24.7%

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

       6. sub-neg 24.7%

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

       7. *-commutative 24.7%

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

   12. Simplified 24.7%

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

   if -0.5 < x < 500

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in x around 0 70.1%

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

   if 500 < x < 7.5000000000000001e217

   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 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

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

      3. rec-exp 55.1%

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

      4. sub-neg 55.1%

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

      5. div-sub 55.1%

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

      6. mul-1-neg 55.1%

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

      7. rec-exp 55.1%

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

      8. +-inverses 55.1%

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

   7. Simplified 55.1%

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

   if 7.5000000000000001e217 < 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 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 39.1%

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

   7. Taylor expanded in eps around inf 35.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 620:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot 2}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.5)
   (/ (* x (- -1.0 eps)) 2.0)
   (if (<= x 620.0)
     (/ (/ (* eps 2.0) eps) 2.0)
     (if (<= x 7.5e+217) 0.0 (/ (* x eps) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.5) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 620.0) {
		tmp = ((eps * 2.0) / eps) / 2.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.5d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else if (x <= 620.0d0) then
        tmp = ((eps * 2.0d0) / eps) / 2.0d0
    else if (x <= 7.5d+217) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.5) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 620.0) {
		tmp = ((eps * 2.0) / eps) / 2.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.5:
		tmp = (x * (-1.0 - eps)) / 2.0
	elif x <= 620.0:
		tmp = ((eps * 2.0) / eps) / 2.0
	elif x <= 7.5e+217:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.5)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	elseif (x <= 620.0)
		tmp = Float64(Float64(Float64(eps * 2.0) / eps) / 2.0);
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.5)
		tmp = (x * (-1.0 - eps)) / 2.0;
	elseif (x <= 620.0)
		tmp = ((eps * 2.0) / eps) / 2.0;
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.5], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 620.0], N[(N[(N[(eps * 2.0), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.5e+217], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.5

   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 58.5%

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

   6. Taylor expanded in eps around inf 58.5%

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

      1. cancel-sign-sub-inv 58.5%

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

      2. metadata-eval 58.5%

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

      3. *-lft-identity 58.5%

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

      4. associate-*r* 58.5%

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

      5. +-commutative 58.5%

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

      6. exp-prod 58.5%

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

      7. +-commutative 58.5%

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

      8. *-lft-identity 58.5%

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

      9. metadata-eval 58.5%

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

      10. cancel-sign-sub-inv 58.5%

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

      11. exp-prod 58.5%

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

      12. associate-*r* 58.5%

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

      13. mul-1-neg 58.5%

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

      14. mul-1-neg 58.5%

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

      15. associate-*r* 58.5%

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

      16. *-commutative 58.5%

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

      17. associate-*l* 58.5%

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

      18. cancel-sign-sub-inv 58.5%

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

      19. metadata-eval 58.5%

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

      20. *-lft-identity 58.5%

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

      21. distribute-lft-in 58.5%

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

   8. Simplified 58.5%

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

   9. Taylor expanded in x around 0 24.7%

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

   10. Taylor expanded in x around inf 24.7%

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

       1. *-commutative 24.7%

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

       2. associate-*r* 24.7%

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

       3. distribute-lft-in 24.7%

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

       4. metadata-eval 24.7%

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

       5. neg-mul-1 24.7%

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

       6. sub-neg 24.7%

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

       7. *-commutative 24.7%

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

   12. Simplified 24.7%

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

   if -0.5 < x < 620

   1. Initial program 56.9%

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

   3. Simplified 36.0%

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

   5. Taylor expanded in eps around 0 27.0%

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

   7. Simplified 71.5%

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

   8. Taylor expanded in x around 0 70.8%

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

   if 620 < x < 7.5000000000000001e217

   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 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

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

      3. rec-exp 55.1%

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

      4. sub-neg 55.1%

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

      5. div-sub 55.1%

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

      6. mul-1-neg 55.1%

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

      7. rec-exp 55.1%

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

      8. +-inverses 55.1%

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

   7. Simplified 55.1%

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

   if 7.5000000000000001e217 < 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 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 39.1%

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

   7. Taylor expanded in eps around inf 35.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 620:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot 2}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 210.0)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0)
   (if (<= x 7.6e+217) 0.0 (/ (* x eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 7.6e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 210.0d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    else if (x <= 7.6d+217) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 7.6e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 210.0:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	elif x <= 7.6e+217:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 210.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	elseif (x <= 7.6e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 210.0)
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	elseif (x <= 7.6e+217)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 210.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.6e+217], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 210

   1. Initial program 65.8%

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

   3. Simplified 54.7%

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

   5. Taylor expanded in x around 0 56.3%

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

   6. Taylor expanded in eps around 0 60.1%

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

   if 210 < x < 7.60000000000000004e217

   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 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

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

      3. rec-exp 55.1%

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

      4. sub-neg 55.1%

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

      5. div-sub 55.1%

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

      6. mul-1-neg 55.1%

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

      7. rec-exp 55.1%

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

      8. +-inverses 55.1%

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

   7. Simplified 55.1%

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

   if 7.60000000000000004e217 < 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 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 39.1%

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

   7. Taylor expanded in eps around inf 35.9%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.0% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 1\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 2.0 (* x (+ eps 1.0))) 2.0)
   (if (<= x 7.5e+217) 0.0 (/ (* x eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - (x * (eps + 1.0))) / 2.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = (2.0d0 - (x * (eps + 1.0d0))) / 2.0d0
    else if (x <= 7.5d+217) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - (x * (eps + 1.0))) / 2.0;
	} else if (x <= 7.5e+217) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - (x * (eps + 1.0))) / 2.0
	elif x <= 7.5e+217:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 1.0))) / 2.0);
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - (x * (eps + 1.0))) / 2.0;
	elseif (x <= 7.5e+217)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 - N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.5e+217], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around 0 45.6%

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

   6. Taylor expanded in eps around inf 78.5%

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

      1. cancel-sign-sub-inv 78.5%

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

      2. metadata-eval 78.5%

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

      3. *-lft-identity 78.5%

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

      4. associate-*r* 78.5%

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

      5. +-commutative 78.5%

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

      6. exp-prod 68.7%

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

      7. +-commutative 68.7%

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

      8. *-lft-identity 68.7%

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

      9. metadata-eval 68.7%

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

      10. cancel-sign-sub-inv 68.7%

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

      11. exp-prod 78.5%

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

      12. associate-*r* 78.5%

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

      13. mul-1-neg 78.5%

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

      14. mul-1-neg 78.5%

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

      15. associate-*r* 78.5%

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

      16. *-commutative 78.5%

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

      17. associate-*l* 78.5%

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

      18. cancel-sign-sub-inv 78.5%

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

      19. metadata-eval 78.5%

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

      20. *-lft-identity 78.5%

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

      21. distribute-lft-in 78.5%

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

   8. Simplified 78.5%

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

   9. Taylor expanded in x around 0 59.3%

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

   10. Taylor expanded in x around 0 59.3%

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

       1. associate-*r* 59.3%

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

       2. neg-mul-1 59.3%

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

       3. +-commutative 59.3%

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

   12. Simplified 59.3%

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

   if 2 < x < 7.5000000000000001e217

   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 55.1%

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

      1. mul-1-neg 55.1%

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

      2. mul-1-neg 55.1%

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

      3. rec-exp 55.1%

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

      4. sub-neg 55.1%

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

      5. div-sub 55.1%

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

      6. mul-1-neg 55.1%

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

      7. rec-exp 55.1%

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

      8. +-inverses 55.1%

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

   7. Simplified 55.1%

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

   if 7.5000000000000001e217 < 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 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 39.1%

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

   7. Taylor expanded in eps around inf 35.9%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 1\right)}{2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.0% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 510:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.85) (* (* x eps) -0.5) (if (<= x 510.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.85) {
		tmp = (x * eps) * -0.5;
	} else if (x <= 510.0) {
		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 <= (-0.85d0)) then
        tmp = (x * eps) * (-0.5d0)
    else if (x <= 510.0d0) 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 <= -0.85) {
		tmp = (x * eps) * -0.5;
	} else if (x <= 510.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.85:
		tmp = (x * eps) * -0.5
	elif x <= 510.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.85)
		tmp = Float64(Float64(x * eps) * -0.5);
	elseif (x <= 510.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.85)
		tmp = (x * eps) * -0.5;
	elseif (x <= 510.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.85], N[(N[(x * eps), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 510.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.849999999999999978

   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 66.5%

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

   7. Taylor expanded in eps around inf 32.3%

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

      1. frac-2neg 32.3%

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

      2. div-inv 32.3%

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

      3. distribute-rgt-neg-in 32.3%

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

      4. add-sqr-sqrt 32.3%

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

      5. sqrt-unprod 32.3%

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

      6. sqr-neg 32.3%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 24.7%

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

      9. *-commutative 24.7%

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

      10. metadata-eval 24.7%

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

      11. metadata-eval 24.7%

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

   9. Applied egg-rr 24.7%

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

   if -0.849999999999999978 < x < 510

   1. Initial program 56.9%

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

   3. Simplified 56.9%

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

   5. Taylor expanded in x around 0 70.1%

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

   if 510 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 48.5%

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

      1. mul-1-neg 48.5%

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

      2. mul-1-neg 48.5%

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

      3. rec-exp 48.5%

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

      4. sub-neg 48.5%

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

      5. div-sub 48.5%

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

      6. mul-1-neg 48.5%

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

      7. rec-exp 48.5%

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

      8. +-inverses 48.5%

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

   7. Simplified 48.5%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 510:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 480.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 480.0) {
		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 <= 480.0d0) 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 <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 480.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 480.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 480.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 480

   1. Initial program 65.8%

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

   3. Simplified 65.8%

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

   5. Taylor expanded in x around 0 56.3%

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

   if 480 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 48.5%

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

      1. mul-1-neg 48.5%

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

      2. mul-1-neg 48.5%

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

      3. rec-exp 48.5%

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

      4. sub-neg 48.5%

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

      5. div-sub 48.5%

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

      6. mul-1-neg 48.5%

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

      7. rec-exp 48.5%

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

      8. +-inverses 48.5%

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

   7. Simplified 48.5%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 77.3%

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

3. Simplified 66.2%

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

5. Taylor expanded in eps around 0 17.7%

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

   1. mul-1-neg 17.7%

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

   2. mul-1-neg 17.7%

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

   3. rec-exp 17.7%

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

   4. sub-neg 17.7%

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

   5. div-sub 17.7%

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

   6. mul-1-neg 17.7%

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

   7. rec-exp 17.7%

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

   8. +-inverses 17.9%

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

7. Simplified 17.9%

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

8. Final simplification 17.9%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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))