NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.0% → 99.9%

Time: 13.7s

Alternatives: 16

Speedup: 0.4×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.49737811306551e+57
  2. eps = -8.221534230581209e-277

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 2:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\varepsilon \cdot x} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow eps -1.0))))
   (if (<=
        (-
         (* t_0 (exp (* (+ -1.0 eps) x)))
         (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
        2.0)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (/ (- (* t_0 (exp (* eps x))) (- (exp (- (fma x eps x))))) 2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (((t_0 * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 2.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = ((t_0 * exp((eps * x))) - -exp(-fma(x, eps, x))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 2.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(eps * x))) - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2

   1. Initial program 68.7%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 2 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 2:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\varepsilon \cdot x} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\varepsilon \cdot x} - -1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow eps -1.0))))
   (if (<=
        (-
         (* t_0 (exp (* (+ -1.0 eps) x)))
         (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
        4.0)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (/ (- (* t_0 (exp (* eps x))) -1.0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (((t_0 * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = ((t_0 * exp((eps * x))) - -1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (eps ** (-1.0d0))
    if (((t_0 * exp((((-1.0d0) + eps) * x))) - (((eps ** (-1.0d0)) - 1.0d0) * exp((((-1.0d0) - eps) * x)))) <= 4.0d0) then
        tmp = (2.0d0 * ((1.0d0 + x) / exp(x))) * 0.5d0
    else
        tmp = ((t_0 * exp((eps * x))) - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 + Math.pow(eps, -1.0);
	double tmp;
	if (((t_0 * Math.exp(((-1.0 + eps) * x))) - ((Math.pow(eps, -1.0) - 1.0) * Math.exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / Math.exp(x))) * 0.5;
	} else {
		tmp = ((t_0 * Math.exp((eps * x))) - -1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 + math.pow(eps, -1.0)
	tmp = 0
	if ((t_0 * math.exp(((-1.0 + eps) * x))) - ((math.pow(eps, -1.0) - 1.0) * math.exp(((-1.0 - eps) * x)))) <= 4.0:
		tmp = (2.0 * ((1.0 + x) / math.exp(x))) * 0.5
	else:
		tmp = ((t_0 * math.exp((eps * x))) - -1.0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(eps * x))) - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 + (eps ^ -1.0);
	tmp = 0.0;
	if (((t_0 * exp(((-1.0 + eps) * x))) - (((eps ^ -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0)
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	else
		tmp = ((t_0 * exp((eps * x))) - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 68.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 56.0

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

   11. Applied rewrites 56.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\varepsilon \cdot x} - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      4.0)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (/ (- (+ (pow eps -1.0) 1.0) (- (exp (- (fma x eps x))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = ((pow(eps, -1.0) + 1.0) - -exp(-fma(x, eps, x))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) + 1.0) - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 68.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 47.1

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

   10. Applied rewrites 47.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+61} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow eps -1.0))))
   (if (<= eps 1.0)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (if (or (<= eps 1.6e+61) (not (<= eps 2.3e+279)))
       (/
        (-
         (* t_0 (fma (- eps 1.0) x 1.0))
         (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
        2.0)
       (/
        (+
         (* t_0 (exp (* (+ -1.0 eps) x)))
         (* (- (/ -1.0 eps) -1.0) (fma (- -1.0 eps) x 1.0)))
        2.0)))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (eps <= 1.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if ((eps <= 1.6e+61) || !(eps <= 2.3e+279)) {
		tmp = ((t_0 * fma((eps - 1.0), x, 1.0)) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	} else {
		tmp = ((t_0 * exp(((-1.0 + eps) * x))) + (((-1.0 / eps) - -1.0) * fma((-1.0 - eps), x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif ((eps <= 1.6e+61) || !(eps <= 2.3e+279))
		tmp = Float64(Float64(Float64(t_0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) + Float64(Float64(Float64(-1.0 / eps) - -1.0) * fma(Float64(-1.0 - eps), x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[eps, 1.6e+61], N[Not[LessEqual[eps, 2.3e+279]], $MachinePrecision]], N[(N[(N[(t$95$0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 74.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 1 < eps < 1.5999999999999999e61 or 2.3e279 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 1.5999999999999999e61 < eps < 2.3e279

   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

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 70.2

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

   5. Applied rewrites 70.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+61} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+61} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\varepsilon \cdot x} - \mathsf{fma}\left(1 + \varepsilon, x, -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (pow eps -1.0))))
   (if (<= eps 1.0)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (if (or (<= eps 1.6e+61) (not (<= eps 2.3e+279)))
       (/
        (-
         (* t_0 (fma (- eps 1.0) x 1.0))
         (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
        2.0)
       (/ (- (* t_0 (exp (* eps x))) (fma (+ 1.0 eps) x -1.0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + pow(eps, -1.0);
	double tmp;
	if (eps <= 1.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if ((eps <= 1.6e+61) || !(eps <= 2.3e+279)) {
		tmp = ((t_0 * fma((eps - 1.0), x, 1.0)) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	} else {
		tmp = ((t_0 * exp((eps * x))) - fma((1.0 + eps), x, -1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif ((eps <= 1.6e+61) || !(eps <= 2.3e+279))
		tmp = Float64(Float64(Float64(t_0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(eps * x))) - fma(Float64(1.0 + eps), x, -1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, 1.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[eps, 1.6e+61], N[Not[LessEqual[eps, 2.3e+279]], $MachinePrecision]], N[(N[(N[(t$95$0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + eps), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 74.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 1 < eps < 1.5999999999999999e61 or 2.3e279 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 1.5999999999999999e61 < eps < 2.3e279

   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 eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 70.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{+61} \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{+279}\right):\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\varepsilon \cdot x} - \mathsf{fma}\left(1 + \varepsilon, x, -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -23000000000000:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -23000000000000.0)
     (/ (- (/ (exp (- x)) eps) -1.0) 2.0)
     (if (<= x -2.3e-246)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5e+274)
         (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
         (fma (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -23000000000000.0) {
		tmp = ((exp(-x) / eps) - -1.0) / 2.0;
	} else if (x <= -2.3e-246) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5e+274) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -23000000000000.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps) - -1.0) / 2.0);
	elseif (x <= -2.3e-246)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 5e+274)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -23000000000000.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2.3e-246], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5e+274], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.3e13

   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 eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.0%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. neg-mul-1 N/A

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

       3. lower-exp.f64 N/A

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

       4. neg-mul-1 N/A

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

       5. lower-neg.f64 36.1

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

   11. Applied rewrites 36.1%

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

   if -2.3e13 < x < -2.2999999999999998e-246

   1. Initial program 68.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 84.2%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right) \]

   if -2.2999999999999998e-246 < x < 4.9999999999999998e274

   1. Initial program 84.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 4.9999999999999998e274 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 71.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \frac{1}{\varepsilon}\right), 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -23000000000000:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-246}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8200:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{-x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\varepsilon - 1\right) + \left(-\varepsilon\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 8200.0)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (if (<= eps 6e+210)
     (/ (- (* (+ 1.0 (pow eps -1.0)) (exp (- x))) -1.0) 2.0)
     (fma
      (* 0.5 x)
      (+ (/ (/ (fma eps eps -1.0) (- eps -1.0)) eps) (+ (- eps 1.0) (- eps)))
      1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 8200.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if (eps <= 6e+210) {
		tmp = (((1.0 + pow(eps, -1.0)) * exp(-x)) - -1.0) / 2.0;
	} else {
		tmp = fma((0.5 * x), (((fma(eps, eps, -1.0) / (eps - -1.0)) / eps) + ((eps - 1.0) + -eps)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 8200.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif (eps <= 6e+210)
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(-x))) - -1.0) / 2.0);
	else
		tmp = fma(Float64(0.5 * x), Float64(Float64(Float64(fma(eps, eps, -1.0) / Float64(eps - -1.0)) / eps) + Float64(Float64(eps - 1.0) + Float64(-eps))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 8200.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 6e+210], N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(N[(N[(eps * eps + -1.0), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + N[(N[(eps - 1.0), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 8200

   1. Initial program 75.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 8200 < eps < 6.00000000000000044e210

   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 eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.3%

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

   9. Taylor expanded in eps around 0

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 53.6

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

   11. Applied rewrites 53.6%

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

   if 6.00000000000000044e210 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 7.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 7.2%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 7.5%

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

   12. Applied rewrites 58.5%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8200:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{-x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\varepsilon - 1\right) + \left(-\varepsilon\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\varepsilon - 1\right) + \left(-\varepsilon\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.0)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (if (<= eps 9.5e+179)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (fma
      (* 0.5 x)
      (+ (/ (/ (fma eps eps -1.0) (- eps -1.0)) eps) (+ (- eps 1.0) (- eps)))
      1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if (eps <= 9.5e+179) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = fma((0.5 * x), (((fma(eps, eps, -1.0) / (eps - -1.0)) / eps) + ((eps - 1.0) + -eps)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif (eps <= 9.5e+179)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = fma(Float64(0.5 * x), Float64(Float64(Float64(fma(eps, eps, -1.0) / Float64(eps - -1.0)) / eps) + Float64(Float64(eps - 1.0) + Float64(-eps))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 9.5e+179], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(N[(N[(eps * eps + -1.0), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + N[(N[(eps - 1.0), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 74.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   if 1 < eps < 9.5e179

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 56.0%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   if 9.5e179 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 11.7%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 11.7%

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

   12. Applied rewrites 47.2%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\color{blue}{\varepsilon} - 1\right) + \left(-\varepsilon\right)\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;\left(x + 1\right) \cdot e^{-x}\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\varepsilon - 1\right) + \left(-\varepsilon\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.0)
   (* (+ x 1.0) (exp (- x)))
   (if (<= eps 9.5e+179)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (fma
      (* 0.5 x)
      (+ (/ (/ (fma eps eps -1.0) (- eps -1.0)) eps) (+ (- eps 1.0) (- eps)))
      1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = (x + 1.0) * exp(-x);
	} else if (eps <= 9.5e+179) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = fma((0.5 * x), (((fma(eps, eps, -1.0) / (eps - -1.0)) / eps) + ((eps - 1.0) + -eps)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(-x)));
	elseif (eps <= 9.5e+179)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = fma(Float64(0.5 * x), Float64(Float64(Float64(fma(eps, eps, -1.0) / Float64(eps - -1.0)) / eps) + Float64(Float64(eps - 1.0) + Float64(-eps))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.0], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 9.5e+179], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(N[(N[(eps * eps + -1.0), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + N[(N[(eps - 1.0), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 74.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

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

   if 1 < eps < 9.5e179

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 56.0%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   if 9.5e179 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 11.7%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 11.7%

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

   12. Applied rewrites 47.2%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\color{blue}{\varepsilon} - 1\right) + \left(-\varepsilon\right)\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1020000000:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 9.5 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\varepsilon - 1\right) + \left(-\varepsilon\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1020000000.0)
   (*
    (*
     2.0
     (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)))
    0.5)
   (if (<= eps 9.5e+179)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (fma
      (* 0.5 x)
      (+ (/ (/ (fma eps eps -1.0) (- eps -1.0)) eps) (+ (- eps 1.0) (- eps)))
      1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1020000000.0) {
		tmp = (2.0 * ((1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5;
	} else if (eps <= 9.5e+179) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = fma((0.5 * x), (((fma(eps, eps, -1.0) / (eps - -1.0)) / eps) + ((eps - 1.0) + -eps)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1020000000.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5);
	elseif (eps <= 9.5e+179)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = fma(Float64(0.5 * x), Float64(Float64(Float64(fma(eps, eps, -1.0) / Float64(eps - -1.0)) / eps) + Float64(Float64(eps - 1.0) + Float64(-eps))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1020000000.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 9.5e+179], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(N[(N[(eps * eps + -1.0), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] + N[(N[(eps - 1.0), $MachinePrecision] + (-eps)), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.02e9

   1. Initial program 75.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \frac{1 + x}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.9%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]

   if 1.02e9 < eps < 9.5e179

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 55.4%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   if 9.5e179 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 11.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 11.7%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 11.7%

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

   12. Applied rewrites 47.2%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \frac{\frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - -1}}{\varepsilon} + \left(\left(\color{blue}{\varepsilon} - 1\right) + \left(-\varepsilon\right)\right), 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1020000000:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.3 \cdot 10^{+235}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon - 1, \left(\frac{x}{\varepsilon} + x\right) \cdot 0.5, \mathsf{fma}\left(0.5 \cdot x, -\varepsilon, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1020000000.0)
   (*
    (*
     2.0
     (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)))
    0.5)
   (if (<= eps 1.3e+235)
     (fma (* 0.3333333333333333 x) (* x x) 1.0)
     (fma (- eps 1.0) (* (+ (/ x eps) x) 0.5) (fma (* 0.5 x) (- eps) 1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1020000000.0) {
		tmp = (2.0 * ((1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5;
	} else if (eps <= 1.3e+235) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = fma((eps - 1.0), (((x / eps) + x) * 0.5), fma((0.5 * x), -eps, 1.0));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1020000000.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5);
	elseif (eps <= 1.3e+235)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = fma(Float64(eps - 1.0), Float64(Float64(Float64(x / eps) + x) * 0.5), fma(Float64(0.5 * x), Float64(-eps), 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1020000000.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 1.3e+235], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(eps - 1.0), $MachinePrecision] * N[(N[(N[(x / eps), $MachinePrecision] + x), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[(0.5 * x), $MachinePrecision] * (-eps) + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.02e9

   1. Initial program 75.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(2 \cdot \frac{1 + x}{1 + x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)}\right) \cdot \frac{1}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.9%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]

   if 1.02e9 < eps < 1.2999999999999999e235

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 50.5%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   if 1.2999999999999999e235 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 7.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 7.7%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 20.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 20.4%

       \[\leadsto \mathsf{fma}\left(\varepsilon - 1, \left(\frac{x}{\varepsilon} + x\right) \cdot \color{blue}{0.5}, \mathsf{fma}\left(0.5 \cdot x, -\varepsilon, 1\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.2% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -215:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon - 1, \frac{x}{\varepsilon} \cdot 0.5, \mathsf{fma}\left(0.5 \cdot x, -\varepsilon, 1\right)\right)\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+274}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -215.0)
   (fma (- eps 1.0) (* (/ x eps) 0.5) (fma (* 0.5 x) (- eps) 1.0))
   (if (<= x 1.8)
     (fma (fma (fma -0.125 x 0.3333333333333333) x -0.5) (* x x) 1.0)
     (if (<= x 5e+274) 0.0 (fma (* 0.3333333333333333 x) (* x x) 1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -215.0) {
		tmp = fma((eps - 1.0), ((x / eps) * 0.5), fma((0.5 * x), -eps, 1.0));
	} else if (x <= 1.8) {
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), (x * x), 1.0);
	} else if (x <= 5e+274) {
		tmp = 0.0;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -215.0)
		tmp = fma(Float64(eps - 1.0), Float64(Float64(x / eps) * 0.5), fma(Float64(0.5 * x), Float64(-eps), 1.0));
	elseif (x <= 1.8)
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), Float64(x * x), 1.0);
	elseif (x <= 5e+274)
		tmp = 0.0;
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -215.0], N[(N[(eps - 1.0), $MachinePrecision] * N[(N[(x / eps), $MachinePrecision] * 0.5), $MachinePrecision] + N[(N[(0.5 * x), $MachinePrecision] * (-eps) + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8], N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5e+274], 0.0, N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -215

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 3.1%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 21.9%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 21.4%

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

   if -215 < x < 1.80000000000000004

   1. Initial program 68.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   if 1.80000000000000004 < x < 4.9999999999999998e274

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 2.4%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.9%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 51.5%

       \[\leadsto 0 \]

   if 4.9999999999999998e274 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 15.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 85.9%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.3% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1020000000:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1020000000.0)
   (* (* 2.0 (/ (+ 1.0 x) (fma (fma 0.5 x 1.0) x 1.0))) 0.5)
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1020000000.0) {
		tmp = (2.0 * ((1.0 + x) / fma(fma(0.5, x, 1.0), x, 1.0))) * 0.5;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1020000000.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / fma(fma(0.5, x, 1.0), x, 1.0))) * 0.5);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1020000000.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.02e9

   1. Initial program 75.4%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.8%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\right) \cdot 0.5 \]

   if 1.02e9 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 27.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 44.3%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+274}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 550.0)
   1.0
   (if (<= x 5e+274) 0.0 (fma (* 0.3333333333333333 x) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 5e+274) {
		tmp = 0.0;
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 5e+274)
		tmp = 0.0;
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, If[LessEqual[x, 5e+274], 0.0, N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 550

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{1} \]

   if 550 < x < 4.9999999999999998e274

   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

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 2.4%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.9%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 51.5%

       \[\leadsto 0 \]

   if 4.9999999999999998e274 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 15.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 85.9%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 56.7% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		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 <= 550.0d0) 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 <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 75.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.8%

      \[\leadsto \color{blue}{1} \]

   if 550 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 2.4%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, -\varepsilon\right), 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 14.0%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 48.3%

       \[\leadsto 0 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.0% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 83.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 40.0%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024314 
(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))