NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.6% → 98.8%

Time: 20.7s

Alternatives: 13

Speedup: 2.0×

• 38.0% 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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + x \cdot \varepsilon}}\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (+ (exp (* x (+ eps -1.0))) (/ 1.0 (exp (+ x (* x eps))))) 0.5))

C

double code(double x, double eps) {
	return (exp((x * (eps + -1.0))) + (1.0 / exp((x + (x * eps))))) * 0.5;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp((x * (eps + (-1.0d0)))) + (1.0d0 / exp((x + (x * eps))))) * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return (Math.exp((x * (eps + -1.0))) + (1.0 / Math.exp((x + (x * eps))))) * 0.5;
}

Python

def code(x, eps):
	return (math.exp((x * (eps + -1.0))) + (1.0 / math.exp((x + (x * eps))))) * 0.5

Julia

function code(x, eps)
	return Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + Float64(1.0 / exp(Float64(x + Float64(x * eps))))) * 0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = (exp((x * (eps + -1.0))) + (1.0 / exp((x + (x * eps))))) * 0.5;
end

Wolfram

code[x_, eps_] := N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(x * eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Simplified 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
4. Add Preprocessing

5. Taylor expanded in eps around inf 98.6%

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

6. Final simplification 98.6%

   \[\leadsto \left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{e^{x + x \cdot \varepsilon}}\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 2: 84.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \varepsilon}\\ \mathbf{if}\;x \leq 21500:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \frac{1}{t\_0}\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+92} \lor \neg \left(x \leq 8.6 \cdot 10^{+150}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x eps))))
   (if (<= x 21500.0)
     (* 0.5 (+ t_0 (/ 1.0 t_0)))
     (if (or (<= x 2.4e+92) (not (<= x 8.6e+150)))
       0.0
       (*
        0.5
        (+ (exp (* x (+ eps -1.0))) (/ 1.0 (+ 1.0 (* x (+ eps 1.0))))))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * eps));
	double tmp;
	if (x <= 21500.0) {
		tmp = 0.5 * (t_0 + (1.0 / t_0));
	} else if ((x <= 2.4e+92) || !(x <= 8.6e+150)) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * eps))
    if (x <= 21500.0d0) then
        tmp = 0.5d0 * (t_0 + (1.0d0 / t_0))
    else if ((x <= 2.4d+92) .or. (.not. (x <= 8.6d+150))) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * (exp((x * (eps + (-1.0d0)))) + (1.0d0 / (1.0d0 + (x * (eps + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * eps));
	double tmp;
	if (x <= 21500.0) {
		tmp = 0.5 * (t_0 + (1.0 / t_0));
	} else if ((x <= 2.4e+92) || !(x <= 8.6e+150)) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (Math.exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * eps))
	tmp = 0
	if x <= 21500.0:
		tmp = 0.5 * (t_0 + (1.0 / t_0))
	elif (x <= 2.4e+92) or not (x <= 8.6e+150):
		tmp = 0.0
	else:
		tmp = 0.5 * (math.exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))))
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * eps))
	tmp = 0.0
	if (x <= 21500.0)
		tmp = Float64(0.5 * Float64(t_0 + Float64(1.0 / t_0)));
	elseif ((x <= 2.4e+92) || !(x <= 8.6e+150))
		tmp = 0.0;
	else
		tmp = Float64(0.5 * Float64(exp(Float64(x * Float64(eps + -1.0))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps + 1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * eps));
	tmp = 0.0;
	if (x <= 21500.0)
		tmp = 0.5 * (t_0 + (1.0 / t_0));
	elseif ((x <= 2.4e+92) || ~((x <= 8.6e+150)))
		tmp = 0.0;
	else
		tmp = 0.5 * (exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 21500.0], N[(0.5 * N[(t$95$0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 2.4e+92], N[Not[LessEqual[x, 8.6e+150]], $MachinePrecision]], 0.0, N[(0.5 * N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 21500

   1. Initial program 63.9%

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

   3. Simplified 55.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 98.1%

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

   6. Taylor expanded in eps around inf 98.2%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

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

   8. Simplified 98.2%

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

   9. Taylor expanded in eps around inf 98.6%

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

   if 21500 < x < 2.40000000000000005e92 or 8.59999999999999994e150 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 64.0%

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

      1. rec-exp 64.0%

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

      2. div-sub 64.0%

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

      3. neg-mul-1 64.0%

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

      4. +-inverses 64.0%

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

   7. Simplified 64.0%

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

   if 2.40000000000000005e92 < x < 8.59999999999999994e150

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 57.8%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 21500:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \varepsilon} + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+92} \lor \neg \left(x \leq 8.6 \cdot 10^{+150}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 33000000 \lor \neg \left(x \leq 4.3 \cdot 10^{+89}\right) \land x \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-260)
   (* 0.5 (+ 1.0 (/ 1.0 (exp (* x eps)))))
   (if (or (<= x 33000000.0) (and (not (<= x 4.3e+89)) (<= x 1.85e+151)))
     (* 0.5 (+ (exp (* x (+ eps -1.0))) (/ 1.0 (+ 1.0 (* x (+ eps 1.0))))))
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-260) {
		tmp = 0.5 * (1.0 + (1.0 / exp((x * eps))));
	} else if ((x <= 33000000.0) || (!(x <= 4.3e+89) && (x <= 1.85e+151))) {
		tmp = 0.5 * (exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 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 <= (-4d-260)) then
        tmp = 0.5d0 * (1.0d0 + (1.0d0 / exp((x * eps))))
    else if ((x <= 33000000.0d0) .or. (.not. (x <= 4.3d+89)) .and. (x <= 1.85d+151)) then
        tmp = 0.5d0 * (exp((x * (eps + (-1.0d0)))) + (1.0d0 / (1.0d0 + (x * (eps + 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 <= -4e-260) {
		tmp = 0.5 * (1.0 + (1.0 / Math.exp((x * eps))));
	} else if ((x <= 33000000.0) || (!(x <= 4.3e+89) && (x <= 1.85e+151))) {
		tmp = 0.5 * (Math.exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4e-260:
		tmp = 0.5 * (1.0 + (1.0 / math.exp((x * eps))))
	elif (x <= 33000000.0) or (not (x <= 4.3e+89) and (x <= 1.85e+151)):
		tmp = 0.5 * (math.exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-260)
		tmp = Float64(0.5 * Float64(1.0 + Float64(1.0 / exp(Float64(x * eps)))));
	elseif ((x <= 33000000.0) || (!(x <= 4.3e+89) && (x <= 1.85e+151)))
		tmp = Float64(0.5 * Float64(exp(Float64(x * Float64(eps + -1.0))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps + 1.0))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4e-260)
		tmp = 0.5 * (1.0 + (1.0 / exp((x * eps))));
	elseif ((x <= 33000000.0) || (~((x <= 4.3e+89)) && (x <= 1.85e+151)))
		tmp = 0.5 * (exp((x * (eps + -1.0))) + (1.0 / (1.0 + (x * (eps + 1.0)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-260], N[(0.5 * N[(1.0 + N[(1.0 / N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 33000000.0], And[N[Not[LessEqual[x, 4.3e+89]], $MachinePrecision], LessEqual[x, 1.85e+151]]], N[(0.5 * N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999985e-260

   1. Initial program 69.5%

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

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 96.7%

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

   6. Taylor expanded in eps around inf 96.7%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in eps around inf 97.4%

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

   10. Taylor expanded in x around 0 74.3%

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

   if -3.99999999999999985e-260 < x < 3.3e7 or 4.3000000000000002e89 < x < 1.8499999999999999e151

   1. Initial program 63.8%

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

   3. Simplified 56.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 80.4%

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

   if 3.3e7 < x < 4.3000000000000002e89 or 1.8499999999999999e151 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 64.0%

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

      1. rec-exp 64.0%

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

      2. div-sub 64.0%

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

      3. neg-mul-1 64.0%

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

      4. +-inverses 64.0%

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

   7. Simplified 64.0%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 33000000 \lor \neg \left(x \leq 4.3 \cdot 10^{+89}\right) \land x \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \varepsilon}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{t\_0}\right)\\ \mathbf{elif}\;x \leq 2100000:\\ \;\;\;\;0.5 \cdot \left(t\_0 + \left(1 - x \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+89} \lor \neg \left(x \leq 4.5 \cdot 10^{+149}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x eps))))
   (if (<= x -1.55e-260)
     (* 0.5 (+ 1.0 (/ 1.0 t_0)))
     (if (<= x 2100000.0)
       (* 0.5 (+ t_0 (- 1.0 (* x eps))))
       (if (or (<= x 2.15e+89) (not (<= x 4.5e+149)))
         0.0
         (* 0.5 (+ 1.0 (exp (* x (+ eps -1.0))))))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * eps));
	double tmp;
	if (x <= -1.55e-260) {
		tmp = 0.5 * (1.0 + (1.0 / t_0));
	} else if (x <= 2100000.0) {
		tmp = 0.5 * (t_0 + (1.0 - (x * eps)));
	} else if ((x <= 2.15e+89) || !(x <= 4.5e+149)) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (1.0 + exp((x * (eps + -1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * eps))
    if (x <= (-1.55d-260)) then
        tmp = 0.5d0 * (1.0d0 + (1.0d0 / t_0))
    else if (x <= 2100000.0d0) then
        tmp = 0.5d0 * (t_0 + (1.0d0 - (x * eps)))
    else if ((x <= 2.15d+89) .or. (.not. (x <= 4.5d+149))) then
        tmp = 0.0d0
    else
        tmp = 0.5d0 * (1.0d0 + exp((x * (eps + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * eps));
	double tmp;
	if (x <= -1.55e-260) {
		tmp = 0.5 * (1.0 + (1.0 / t_0));
	} else if (x <= 2100000.0) {
		tmp = 0.5 * (t_0 + (1.0 - (x * eps)));
	} else if ((x <= 2.15e+89) || !(x <= 4.5e+149)) {
		tmp = 0.0;
	} else {
		tmp = 0.5 * (1.0 + Math.exp((x * (eps + -1.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * eps))
	tmp = 0
	if x <= -1.55e-260:
		tmp = 0.5 * (1.0 + (1.0 / t_0))
	elif x <= 2100000.0:
		tmp = 0.5 * (t_0 + (1.0 - (x * eps)))
	elif (x <= 2.15e+89) or not (x <= 4.5e+149):
		tmp = 0.0
	else:
		tmp = 0.5 * (1.0 + math.exp((x * (eps + -1.0))))
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * eps))
	tmp = 0.0
	if (x <= -1.55e-260)
		tmp = Float64(0.5 * Float64(1.0 + Float64(1.0 / t_0)));
	elseif (x <= 2100000.0)
		tmp = Float64(0.5 * Float64(t_0 + Float64(1.0 - Float64(x * eps))));
	elseif ((x <= 2.15e+89) || !(x <= 4.5e+149))
		tmp = 0.0;
	else
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * eps));
	tmp = 0.0;
	if (x <= -1.55e-260)
		tmp = 0.5 * (1.0 + (1.0 / t_0));
	elseif (x <= 2100000.0)
		tmp = 0.5 * (t_0 + (1.0 - (x * eps)));
	elseif ((x <= 2.15e+89) || ~((x <= 4.5e+149)))
		tmp = 0.0;
	else
		tmp = 0.5 * (1.0 + exp((x * (eps + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.55e-260], N[(0.5 * N[(1.0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2100000.0], N[(0.5 * N[(t$95$0 + N[(1.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 2.15e+89], N[Not[LessEqual[x, 4.5e+149]], $MachinePrecision]], 0.0, N[(0.5 * N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.54999999999999991e-260

   1. Initial program 69.5%

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

   3. Simplified 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 96.7%

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

   6. Taylor expanded in eps around inf 96.7%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in eps around inf 97.4%

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

   10. Taylor expanded in x around 0 74.3%

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

   if -1.54999999999999991e-260 < x < 2.1e6

   1. Initial program 56.8%

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

   3. Simplified 47.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

   10. Taylor expanded in eps around 0 85.2%

       \[\leadsto \left(e^{x \cdot \varepsilon} + \color{blue}{\left(1 + -1 \cdot \left(\varepsilon \cdot x\right)\right)}\right) \cdot 0.5 \]
   11. Step-by-step derivation

       1. mul-1-neg 85.2%

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

       2. unsub-neg 85.2%

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

   12. Simplified 85.2%

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

   if 2.1e6 < x < 2.1500000000000001e89 or 4.49999999999999982e149 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 64.0%

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

      1. rec-exp 64.0%

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

      2. div-sub 64.0%

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

      3. neg-mul-1 64.0%

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

      4. +-inverses 64.0%

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

   7. Simplified 64.0%

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

   if 2.1500000000000001e89 < x < 4.49999999999999982e149

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 57.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-260}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 2100000:\\ \;\;\;\;0.5 \cdot \left(e^{x \cdot \varepsilon} + \left(1 - x \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{+89} \lor \neg \left(x \leq 4.5 \cdot 10^{+149}\right):\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 5400000 \lor \neg \left(x \leq 1.65 \cdot 10^{+91}\right) \land x \leq 2.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-287)
   (* 0.5 (+ 1.0 (/ 1.0 (exp (* x eps)))))
   (if (or (<= x 5400000.0) (and (not (<= x 1.65e+91)) (<= x 2.7e+151)))
     (* 0.5 (+ 1.0 (exp (* x (+ eps -1.0)))))
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-287) {
		tmp = 0.5 * (1.0 + (1.0 / exp((x * eps))));
	} else if ((x <= 5400000.0) || (!(x <= 1.65e+91) && (x <= 2.7e+151))) {
		tmp = 0.5 * (1.0 + exp((x * (eps + -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 <= (-2d-287)) then
        tmp = 0.5d0 * (1.0d0 + (1.0d0 / exp((x * eps))))
    else if ((x <= 5400000.0d0) .or. (.not. (x <= 1.65d+91)) .and. (x <= 2.7d+151)) then
        tmp = 0.5d0 * (1.0d0 + exp((x * (eps + (-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 <= -2e-287) {
		tmp = 0.5 * (1.0 + (1.0 / Math.exp((x * eps))));
	} else if ((x <= 5400000.0) || (!(x <= 1.65e+91) && (x <= 2.7e+151))) {
		tmp = 0.5 * (1.0 + Math.exp((x * (eps + -1.0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-287:
		tmp = 0.5 * (1.0 + (1.0 / math.exp((x * eps))))
	elif (x <= 5400000.0) or (not (x <= 1.65e+91) and (x <= 2.7e+151)):
		tmp = 0.5 * (1.0 + math.exp((x * (eps + -1.0))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-287)
		tmp = Float64(0.5 * Float64(1.0 + Float64(1.0 / exp(Float64(x * eps)))));
	elseif ((x <= 5400000.0) || (!(x <= 1.65e+91) && (x <= 2.7e+151)))
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-287)
		tmp = 0.5 * (1.0 + (1.0 / exp((x * eps))));
	elseif ((x <= 5400000.0) || (~((x <= 1.65e+91)) && (x <= 2.7e+151)))
		tmp = 0.5 * (1.0 + exp((x * (eps + -1.0))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-287], N[(0.5 * N[(1.0 + N[(1.0 / N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 5400000.0], And[N[Not[LessEqual[x, 1.65e+91]], $MachinePrecision], LessEqual[x, 2.7e+151]]], N[(0.5 * N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.00000000000000004e-287

   1. Initial program 67.4%

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

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 96.9%

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

   6. Taylor expanded in eps around inf 96.9%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in eps around inf 97.6%

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

   10. Taylor expanded in x around 0 75.8%

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

   if -2.00000000000000004e-287 < x < 5.4e6 or 1.65000000000000009e91 < x < 2.7000000000000001e151

   1. Initial program 65.9%

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

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 78.2%

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

   if 5.4e6 < x < 1.65000000000000009e91 or 2.7000000000000001e151 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 64.0%

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

      1. rec-exp 64.0%

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

      2. div-sub 64.0%

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

      3. neg-mul-1 64.0%

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

      4. +-inverses 64.0%

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

   7. Simplified 64.0%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-287}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 5400000 \lor \neg \left(x \leq 1.65 \cdot 10^{+91}\right) \land x \leq 2.7 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \varepsilon}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{t\_0}\right)\\ \mathbf{elif}\;x \leq 15000 \lor \neg \left(x \leq 1.25 \cdot 10^{+88}\right) \land x \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x eps))))
   (if (<= x -1e-291)
     (* 0.5 (+ 1.0 (/ 1.0 t_0)))
     (if (or (<= x 15000.0) (and (not (<= x 1.25e+88)) (<= x 1.05e+150)))
       (* 0.5 (+ 1.0 t_0))
       0.0))))

C

double code(double x, double eps) {
	double t_0 = exp((x * eps));
	double tmp;
	if (x <= -1e-291) {
		tmp = 0.5 * (1.0 + (1.0 / t_0));
	} else if ((x <= 15000.0) || (!(x <= 1.25e+88) && (x <= 1.05e+150))) {
		tmp = 0.5 * (1.0 + t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * eps))
    if (x <= (-1d-291)) then
        tmp = 0.5d0 * (1.0d0 + (1.0d0 / t_0))
    else if ((x <= 15000.0d0) .or. (.not. (x <= 1.25d+88)) .and. (x <= 1.05d+150)) then
        tmp = 0.5d0 * (1.0d0 + t_0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * eps));
	double tmp;
	if (x <= -1e-291) {
		tmp = 0.5 * (1.0 + (1.0 / t_0));
	} else if ((x <= 15000.0) || (!(x <= 1.25e+88) && (x <= 1.05e+150))) {
		tmp = 0.5 * (1.0 + t_0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * eps))
	tmp = 0
	if x <= -1e-291:
		tmp = 0.5 * (1.0 + (1.0 / t_0))
	elif (x <= 15000.0) or (not (x <= 1.25e+88) and (x <= 1.05e+150)):
		tmp = 0.5 * (1.0 + t_0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * eps))
	tmp = 0.0
	if (x <= -1e-291)
		tmp = Float64(0.5 * Float64(1.0 + Float64(1.0 / t_0)));
	elseif ((x <= 15000.0) || (!(x <= 1.25e+88) && (x <= 1.05e+150)))
		tmp = Float64(0.5 * Float64(1.0 + t_0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * eps));
	tmp = 0.0;
	if (x <= -1e-291)
		tmp = 0.5 * (1.0 + (1.0 / t_0));
	elseif ((x <= 15000.0) || (~((x <= 1.25e+88)) && (x <= 1.05e+150)))
		tmp = 0.5 * (1.0 + t_0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1e-291], N[(0.5 * N[(1.0 + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 15000.0], And[N[Not[LessEqual[x, 1.25e+88]], $MachinePrecision], LessEqual[x, 1.05e+150]]], N[(0.5 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999962e-292

   1. Initial program 67.4%

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

   3. Simplified 59.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 96.9%

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

   6. Taylor expanded in eps around inf 96.9%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in eps around inf 97.6%

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

   10. Taylor expanded in x around 0 75.8%

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

   if -9.99999999999999962e-292 < x < 15000 or 1.24999999999999999e88 < x < 1.04999999999999999e150

   1. Initial program 65.9%

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

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 95.7%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 95.7%

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

   8. Simplified 95.7%

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

   9. Taylor expanded in eps around inf 94.6%

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

   10. Taylor expanded in eps around 0 78.2%

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

   if 15000 < x < 1.24999999999999999e88 or 1.04999999999999999e150 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 64.0%

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

      1. rec-exp 64.0%

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

      2. div-sub 64.0%

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

      3. neg-mul-1 64.0%

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

      4. +-inverses 64.0%

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

   7. Simplified 64.0%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \left(1 + \frac{1}{e^{x \cdot \varepsilon}}\right)\\ \mathbf{elif}\;x \leq 15000 \lor \neg \left(x \leq 1.25 \cdot 10^{+88}\right) \land x \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-233}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{elif}\;x \leq 29000000 \lor \neg \left(x \leq 2.7 \cdot 10^{+90}\right) \land x \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.7e-233)
   (* 0.5 (+ 1.0 (exp (- x))))
   (if (or (<= x 29000000.0) (and (not (<= x 2.7e+90)) (<= x 1.05e+150)))
     (* 0.5 (+ 1.0 (exp (* x eps))))
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e-233) {
		tmp = 0.5 * (1.0 + exp(-x));
	} else if ((x <= 29000000.0) || (!(x <= 2.7e+90) && (x <= 1.05e+150))) {
		tmp = 0.5 * (1.0 + exp((x * eps)));
	} 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 <= 3.7d-233) then
        tmp = 0.5d0 * (1.0d0 + exp(-x))
    else if ((x <= 29000000.0d0) .or. (.not. (x <= 2.7d+90)) .and. (x <= 1.05d+150)) then
        tmp = 0.5d0 * (1.0d0 + exp((x * eps)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 3.7e-233) {
		tmp = 0.5 * (1.0 + Math.exp(-x));
	} else if ((x <= 29000000.0) || (!(x <= 2.7e+90) && (x <= 1.05e+150))) {
		tmp = 0.5 * (1.0 + Math.exp((x * eps)));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 3.7e-233:
		tmp = 0.5 * (1.0 + math.exp(-x))
	elif (x <= 29000000.0) or (not (x <= 2.7e+90) and (x <= 1.05e+150)):
		tmp = 0.5 * (1.0 + math.exp((x * eps)))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.7e-233)
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(-x))));
	elseif ((x <= 29000000.0) || (!(x <= 2.7e+90) && (x <= 1.05e+150)))
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(x * eps))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 3.7e-233)
		tmp = 0.5 * (1.0 + exp(-x));
	elseif ((x <= 29000000.0) || (~((x <= 2.7e+90)) && (x <= 1.05e+150)))
		tmp = 0.5 * (1.0 + exp((x * eps)));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.7e-233], N[(0.5 * N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 29000000.0], And[N[Not[LessEqual[x, 2.7e+90]], $MachinePrecision], LessEqual[x, 1.05e+150]]], N[(0.5 * N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 3.6999999999999998e-233

   1. Initial program 65.9%

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

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 97.2%

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

   6. Taylor expanded in x around 0 72.4%

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

   7. Taylor expanded in eps around 0 84.5%

      \[\leadsto \left(\color{blue}{e^{-1 \cdot x}} + 1\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. neg-mul-1 84.5%

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

   9. Simplified 84.5%

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

   if 3.6999999999999998e-233 < x < 2.9e7 or 2.7e90 < x < 1.04999999999999999e150

   1. Initial program 68.1%

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

   3. Simplified 59.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 95.1%

      \[\leadsto \left(e^{\color{blue}{\varepsilon \cdot x}} + \frac{1}{e^{x + \varepsilon \cdot x}}\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in eps around inf 93.8%

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

   10. Taylor expanded in eps around 0 75.6%

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

   if 2.9e7 < x < 2.7e90 or 1.04999999999999999e150 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 64.0%

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

      1. rec-exp 64.0%

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

      2. div-sub 64.0%

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

      3. neg-mul-1 64.0%

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

      4. +-inverses 64.0%

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

   7. Simplified 64.0%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-233}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{elif}\;x \leq 29000000 \lor \neg \left(x \leq 2.7 \cdot 10^{+90}\right) \land x \leq 1.05 \cdot 10^{+150}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{x \cdot \varepsilon}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-231}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{elif}\;x \leq 740:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{1 + x \cdot \left(\varepsilon + 1\right)} + \left(1 + \frac{x \cdot \left(\varepsilon \cdot \varepsilon + -1\right)}{\varepsilon + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1e-231)
   (* 0.5 (+ 1.0 (exp (- x))))
   (if (<= x 740.0)
     (*
      0.5
      (+
       (/ 1.0 (+ 1.0 (* x (+ eps 1.0))))
       (+ 1.0 (/ (* x (+ (* eps eps) -1.0)) (+ eps 1.0)))))
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1e-231) {
		tmp = 0.5 * (1.0 + exp(-x));
	} else if (x <= 740.0) {
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 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 <= 1d-231) then
        tmp = 0.5d0 * (1.0d0 + exp(-x))
    else if (x <= 740.0d0) then
        tmp = 0.5d0 * ((1.0d0 / (1.0d0 + (x * (eps + 1.0d0)))) + (1.0d0 + ((x * ((eps * eps) + (-1.0d0))) / (eps + 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 <= 1e-231) {
		tmp = 0.5 * (1.0 + Math.exp(-x));
	} else if (x <= 740.0) {
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 1.0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1e-231:
		tmp = 0.5 * (1.0 + math.exp(-x))
	elif x <= 740.0:
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 1.0))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1e-231)
		tmp = Float64(0.5 * Float64(1.0 + exp(Float64(-x))));
	elseif (x <= 740.0)
		tmp = Float64(0.5 * Float64(Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps + 1.0)))) + Float64(1.0 + Float64(Float64(x * Float64(Float64(eps * eps) + -1.0)) / Float64(eps + 1.0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1e-231)
		tmp = 0.5 * (1.0 + exp(-x));
	elseif (x <= 740.0)
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 1.0))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1e-231], N[(0.5 * N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 740.0], N[(0.5 * N[(N[(1.0 / N[(1.0 + N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * N[(N[(eps * eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 9.9999999999999999e-232

   1. Initial program 65.9%

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

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 97.2%

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

   6. Taylor expanded in x around 0 72.4%

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

   7. Taylor expanded in eps around 0 84.5%

      \[\leadsto \left(\color{blue}{e^{-1 \cdot x}} + 1\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. neg-mul-1 84.5%

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

   9. Simplified 84.5%

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

   if 9.9999999999999999e-232 < x < 740

   1. Initial program 60.0%

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

   3. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 80.6%

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

   7. Taylor expanded in x around 0 70.1%

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. sub-neg 70.1%

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

      2. metadata-eval 70.1%

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

      3. distribute-rgt-in 70.1%

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

      4. neg-mul-1 70.1%

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

      5. *-commutative 70.1%

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

      6. sub-neg 70.1%

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

   9. Simplified 70.1%

      \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \varepsilon - x\right)\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}\right) \cdot 0.5 \]
   10. Step-by-step derivation

       1. *-commutative 70.1%

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

       2. *-un-lft-identity 70.1%

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

       3. distribute-rgt-out-- 70.1%

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

       4. *-commutative 70.1%

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

       5. flip-- 74.6%

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

       6. +-commutative 74.6%

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

       7. associate-*l/ 74.6%

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

       8. metadata-eval 74.6%

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

       9. sub-neg 74.6%

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

       10. metadata-eval 74.6%

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

       11. +-commutative 74.6%

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

   11. Applied egg-rr 74.6%

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

   if 740 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 56.6%

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

      1. rec-exp 56.6%

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

      2. div-sub 56.6%

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

      3. neg-mul-1 56.6%

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

      4. +-inverses 56.6%

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

   7. Simplified 56.6%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-231}:\\ \;\;\;\;0.5 \cdot \left(1 + e^{-x}\right)\\ \mathbf{elif}\;x \leq 740:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{1 + x \cdot \left(\varepsilon + 1\right)} + \left(1 + \frac{x \cdot \left(\varepsilon \cdot \varepsilon + -1\right)}{\varepsilon + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot x}{\varepsilon} \cdot 0.25\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-231}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 740:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{1 + x \cdot \left(\varepsilon + 1\right)} + \left(1 + \frac{x \cdot \left(\varepsilon \cdot \varepsilon + -1\right)}{\varepsilon + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3e+77)
   (* (/ (* x x) eps) 0.25)
   (if (<= x 5e-231)
     1.0
     (if (<= x 740.0)
       (*
        0.5
        (+
         (/ 1.0 (+ 1.0 (* x (+ eps 1.0))))
         (+ 1.0 (/ (* x (+ (* eps eps) -1.0)) (+ eps 1.0)))))
       0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3e+77) {
		tmp = ((x * x) / eps) * 0.25;
	} else if (x <= 5e-231) {
		tmp = 1.0;
	} else if (x <= 740.0) {
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 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 <= (-3d+77)) then
        tmp = ((x * x) / eps) * 0.25d0
    else if (x <= 5d-231) then
        tmp = 1.0d0
    else if (x <= 740.0d0) then
        tmp = 0.5d0 * ((1.0d0 / (1.0d0 + (x * (eps + 1.0d0)))) + (1.0d0 + ((x * ((eps * eps) + (-1.0d0))) / (eps + 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 <= -3e+77) {
		tmp = ((x * x) / eps) * 0.25;
	} else if (x <= 5e-231) {
		tmp = 1.0;
	} else if (x <= 740.0) {
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 1.0))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3e+77:
		tmp = ((x * x) / eps) * 0.25
	elif x <= 5e-231:
		tmp = 1.0
	elif x <= 740.0:
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 1.0))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3e+77)
		tmp = Float64(Float64(Float64(x * x) / eps) * 0.25);
	elseif (x <= 5e-231)
		tmp = 1.0;
	elseif (x <= 740.0)
		tmp = Float64(0.5 * Float64(Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps + 1.0)))) + Float64(1.0 + Float64(Float64(x * Float64(Float64(eps * eps) + -1.0)) / Float64(eps + 1.0)))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3e+77)
		tmp = ((x * x) / eps) * 0.25;
	elseif (x <= 5e-231)
		tmp = 1.0;
	elseif (x <= 740.0)
		tmp = 0.5 * ((1.0 / (1.0 + (x * (eps + 1.0)))) + (1.0 + ((x * ((eps * eps) + -1.0)) / (eps + 1.0))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3e+77], N[(N[(N[(x * x), $MachinePrecision] / eps), $MachinePrecision] * 0.25), $MachinePrecision], If[LessEqual[x, 5e-231], 1.0, If[LessEqual[x, 740.0], N[(0.5 * N[(N[(1.0 / N[(1.0 + N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(N[(x * N[(N[(eps * eps), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.9999999999999998e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.9%

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

   4. Taylor expanded in eps around 0 60.7%

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

      1. neg-mul-1 60.7%

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

      2. associate--r+ 60.7%

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

      3. sub-neg 60.7%

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

      4. expm1-define 60.7%

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

      5. remove-double-neg 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in x around 0 53.9%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   8. Step-by-step derivation

      1. *-commutative 53.9%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{\varepsilon} \cdot 0.25} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{\varepsilon} \cdot 0.25 \]

   9. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{x \cdot x}{\varepsilon} \cdot 0.25} \]

   if -2.9999999999999998e77 < x < 5.00000000000000023e-231

   1. Initial program 56.0%

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

   3. Taylor expanded in x around 0 70.7%

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

   if 5.00000000000000023e-231 < x < 740

   1. Initial program 60.0%

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

   3. Simplified 49.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 80.6%

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

   7. Taylor expanded in x around 0 70.1%

      \[\leadsto \left(\color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. sub-neg 70.1%

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

      2. metadata-eval 70.1%

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

      3. distribute-rgt-in 70.1%

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

      4. neg-mul-1 70.1%

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

      5. *-commutative 70.1%

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

      6. sub-neg 70.1%

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

   9. Simplified 70.1%

      \[\leadsto \left(\color{blue}{\left(1 + \left(x \cdot \varepsilon - x\right)\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}\right) \cdot 0.5 \]
   10. Step-by-step derivation

       1. *-commutative 70.1%

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

       2. *-un-lft-identity 70.1%

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

       3. distribute-rgt-out-- 70.1%

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

       4. *-commutative 70.1%

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

       5. flip-- 74.6%

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

       6. +-commutative 74.6%

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

       7. associate-*l/ 74.6%

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

       8. metadata-eval 74.6%

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

       9. sub-neg 74.6%

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

       10. metadata-eval 74.6%

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

       11. +-commutative 74.6%

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

   11. Applied egg-rr 74.6%

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

   if 740 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 56.6%

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

      1. rec-exp 56.6%

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

      2. div-sub 56.6%

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

      3. neg-mul-1 56.6%

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

      4. +-inverses 56.6%

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

   7. Simplified 56.6%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot x}{\varepsilon} \cdot 0.25\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-231}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 740:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{1 + x \cdot \left(\varepsilon + 1\right)} + \left(1 + \frac{x \cdot \left(\varepsilon \cdot \varepsilon + -1\right)}{\varepsilon + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.3% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot x}{\varepsilon} \cdot 0.25\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.1e+77) (* (/ (* x x) eps) 0.25) (if (<= x 600.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e+77) {
		tmp = ((x * x) / eps) * 0.25;
	} else if (x <= 600.0) {
		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 <= (-2.1d+77)) then
        tmp = ((x * x) / eps) * 0.25d0
    else if (x <= 600.0d0) 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 <= -2.1e+77) {
		tmp = ((x * x) / eps) * 0.25;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.1e+77:
		tmp = ((x * x) / eps) * 0.25
	elif x <= 600.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.1e+77)
		tmp = Float64(Float64(Float64(x * x) / eps) * 0.25);
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.1e+77)
		tmp = ((x * x) / eps) * 0.25;
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.1e+77], N[(N[(N[(x * x), $MachinePrecision] / eps), $MachinePrecision] * 0.25), $MachinePrecision], If[LessEqual[x, 600.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0999999999999999e77

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.9%

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

   4. Taylor expanded in eps around 0 60.7%

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

      1. neg-mul-1 60.7%

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

      2. associate--r+ 60.7%

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

      3. sub-neg 60.7%

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

      4. expm1-define 60.7%

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

      5. remove-double-neg 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in x around 0 53.9%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   8. Step-by-step derivation

      1. *-commutative 53.9%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{\varepsilon} \cdot 0.25} \]

      2. unpow2 53.9%

         \[\leadsto \frac{\color{blue}{x \cdot x}}{\varepsilon} \cdot 0.25 \]

   9. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{x \cdot x}{\varepsilon} \cdot 0.25} \]

   if -2.0999999999999999e77 < x < 600

   1. Initial program 57.6%

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

   3. Taylor expanded in x around 0 70.5%

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

   if 600 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 56.6%

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

      1. rec-exp 56.6%

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

      2. div-sub 56.6%

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

      3. neg-mul-1 56.6%

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

      4. +-inverses 56.6%

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

   7. Simplified 56.6%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot x}{\varepsilon} \cdot 0.25\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \varepsilon}{-2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0) (/ (* x eps) (- 2.0)) (if (<= x 520.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps) / -2.0;
	} else if (x <= 520.0) {
		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 <= (-1.0d0)) then
        tmp = (x * eps) / -2.0d0
    else if (x <= 520.0d0) 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 <= -1.0) {
		tmp = (x * eps) / -2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps) / -2.0
	elif x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps) / Float64(-2.0));
	elseif (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps) / -2.0;
	elseif (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.0], N[(N[(x * eps), $MachinePrecision] / (-2.0)), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0 51.1%

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

   4. Taylor expanded in eps around inf 30.7%

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

      1. mul-1-neg 30.7%

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

      2. *-commutative 30.7%

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

      3. distribute-rgt-neg-in 30.7%

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

   6. Simplified 30.7%

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

   if -1 < x < 520

   1. Initial program 55.1%

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

   3. Taylor expanded in x around 0 75.5%

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

   if 520 < 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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 56.6%

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

      1. rec-exp 56.6%

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

      2. div-sub 56.6%

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

      3. neg-mul-1 56.6%

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

      4. +-inverses 56.6%

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

   7. Simplified 56.6%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \varepsilon}{-2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 510:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 510.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	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 <= 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 <= 510.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 510.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (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 <= 510.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 510.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 510

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 0 60.4%

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

   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%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
   4. Add Preprocessing

   5. Taylor expanded in eps around 0 56.6%

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

      1. rec-exp 56.6%

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

      2. div-sub 56.6%

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

      3. neg-mul-1 56.6%

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

      4. +-inverses 56.6%

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

   7. Simplified 56.6%

      \[\leadsto \color{blue}{0} \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 510:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 15.9% 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 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 67.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, {\left(e^{x}\right)}^{\left(\varepsilon + -1\right)}, \frac{1 - \frac{1}{\varepsilon}}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right) \cdot 0.5} \]
4. Add Preprocessing

5. Taylor expanded in eps around 0 16.7%

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

   1. rec-exp 16.6%

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

   2. div-sub 16.6%

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

   3. neg-mul-1 16.6%

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

   4. +-inverses 16.9%

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

7. Simplified 16.9%

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

8. Final simplification 16.9%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(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))